Lowering topological entropy over subsets

LO WERING TOPOLOGICAL ENTR OPY O VER SUBSETS WEN HUANG, XIANGDONG YE AND GUOHUA ZHAN G Abstract. Let ( X, T ) b e a top ological dy na mical system (TDS), and h ( T , K ) the top ological entrop y of a subset K of X . ( X, T ) is lower able if for each 0 ≤ h ≤ h ( T , X ) there is a non-empty compact subse t with entropy h ; is her e ditarily lower ab le if each non-empt y co mpact subset is low erable; is her e d itarily u niformly lower ab le if f or eac h n on-e mpty compact subset K a nd each 0 ≤ h ≤ h ( T , K ) there is a non-empt y compact subset K h ⊆ K with h ( T , K h ) = h a nd K h has a t most one limit point. It is shown that each TDS with finite ent ro p y is low era ble, a nd that a TDS ( X, T ) is hereditarily uniformly low era ble if and only if it is asy mptotically h - expansive. 1. Introduction Throughout the pap er, by a top olo gic al dynam i c al system (TDS) ( X , T ) w e mean a compact metric space X a nd a homeomorphism T : X → X ( in f a ct o ur main results hold for con tin uous ma ps, see App endix). Let ( X , T ) b e a TDS. It is an interes ting question, cons idered in [28] firstly , whether f o r an y g iv en 0 ≤ h ≤ h top ( T , X ), there is a factor ( Y , S ) of ( X , T ) with en trop y h . W e remark that the answ er to this question in the measure-theoretical setup is w ell kno wn, but in the top ological setting the answ er is not completely obtained y et. In [28] Sh ub a nd W eiss presen ted an example with infinite en tropy suc h that each its non-trivial fa ctor has infinite en trop y . Moreov er, Lindenstrauss [20] sho w ed that the question ha s an affirmative answ er when X is finite-dimensional; and f o r an extension of non-trivial minimal Z -actions the question has an affirmative answ er if it has zero me an top ological dimension [21] which includes finite- dimensional systems, systems with finite en trop y and uniquely ergo dic systems. F or the definition and prop erties of mean top ological dimension see [22] by Lindenstrauss a nd W eiss. Let ( X , T ) b e a TDS and K ⊆ X . Denote by h ( T , K ) the top ological en tropy of K . In this pap er w e study a question similar to the ab o v e o ne. Namely , w e consider the question if f o r each 0 ≤ h ≤ h ( T , X ) there is a non-empt y compact subset of X with en tr o p y h . W e remark that the question was motiv ated b y [28 , 2 0, 22, 21] and the w ell-kno wn result in fractal geometry [11, 23] whic h states that if K is a Date : O ctober 18 , 20 08. 2000 Mathematics Su bje ct Classific ation. Primar y: 3 7B40, 3 7A35, 3 7B10, 3 7A05. Key wor ds and phr ases. lowerable, hereditarily low erable, her editarily uniformly lo werable, asymptotically h -expansive, principal extension. The authors are supp orted by a gr an t from Ministry o f Education (2005 0358053 ), NSF C and 973 Pro ject (20 06CB8059 03). The firs t a uthor is supp orted b y F ANEDD (Gran t No 200 520) and the third author is suppor ted by NSF C (108010 3 5). 1 2 lo weri ng top olog ical entrop y ov er subsets non-empt y Borel subset contained in R n then for each 0 ≤ h ≤ dim H ( K ) there is a Borel subset K h of K with d im H ( K h ) = h , where d im H ( ∗ ) is the Hausdorff dimension of a subset ∗ of R n . In [30] Y e and Zhang introduced and studied the not ion o f entrop y p oin ts, and sho w ed that for each non-empt y compact subset K there is a coun table compact subset K 1 ⊆ K with h ( T , K 1 ) = h ( T , K ). Moreo v er, the subset can b e chos en suc h that the limit po ints of the subs et has at most one limit p oin t (for details see [30, Remark 5.13]). Inspired by this fact w e hav e the follow ing not io ns. Definition 1.1. Let ( X, T ) b e a TDS. W e sa y that ( X , T ) is (1) lower able if fo r each 0 ≤ h ≤ h ( T , X ) there is a non-empt y compact subset of X with en trop y h ; (2) her e ditarily low er able if eac h non-empty compact subset is low erable, that is, for eac h non-empty compact subset K ⊆ X and eac h 0 ≤ h ≤ h ( T , K ) there is a non-empty compact subset K h of K with entrop y h ; (3) her e ditarily uniform ly lower able ( HUL for short) if for eac h non- empt y com- pact subset K and each 0 ≤ h ≤ h ( T , K ) there is a non-empty compact subset K h ⊆ K suc h tha t h ( T , K h ) = h and K h has at most one limit p oint. So our question can b e divided further in to the f o llo wing questions. Question 1.2. Is any TDS lower able? Question 1.3. Is any TDS her e ditarily lower able? Question 1.4. F or which TDS it is HUL ? W e remark that lo w ering en trop y for factors is not the same as low ering entrop y for subsets. F or example, in [2 0] Lindenstrauss sho w ed that eac h non- trivial factor of ([0 , 1] Z , σ ) has infinite en trop y , where σ is the shift. But since ( { 0 , 1 , . . . , k } Z , σ ) can b e em b edded as a sub-system of ([0 , 1 ] Z , σ ) for an y k ≥ 1, it is clear that ([0 , 1] Z , σ ) is lo w erable in our sense. In this pap er, w e sho w that eac h TDS with finite en trop y is low erable (this is a lso true when w e talk ab out the dimensional en tro py of a s ubset), and that a TDS is HUL iff it is asymptotically h -expansiv e. In particular, eac h HUL TDS has finite en trop y . Moreo v er, a pr incipal extension preserv es the lo w erable, hereditarily low er- able and HUL prop erties. It is not hard to construct examples with infinite en trop y whic h a re hereditarily lo w erable. Thus , there are TDSs whic h are hereditarily lo w- erable but not HUL . In fact, an example with the same prop erty is explored at the end of the pa per, whic h has finite en tropy . The questions remain op en if there are lo w erable but not here ditarily lo w erable ex amples, or there are TDSs with infinite en trop y whic h are not low erable. W e should remark tha t if ([0 , 1] Z , σ ) is hereditarily lo w erable then eac h finite dimensional TDS without p erio dic p oin ts is hereditarily lo w erable (see [21]), and if it is not then it is a low erable TDS with infinite en tropy whic h is not hereditarily low erable. W e also remark that if there exists a TDS whic h is not lo w erable (suc h a TDS, if e xists, must hav e infinite en trop y) then we can obtain a lo w erable TDS with infinite entrop y whic h is not hereditarily lo w erable b y W en Hu ang, Xiangdong Y e and Guoh ua Zhang 3 considering the union of it and ( [0 , 1] Z , σ ). There are also man y other interes ting questions related to the t o pic. The pap er is o rganized as follow s. In section 2 t he definitions o f top ological en trop y and dimensional en trop y of subsets are giv en, and some basic prop erties are discusse d. In the follow ing section t w o dis tribution principles are stated whic h will b e used in section 4, where it is shown that eac h TDS with finite entrop y is lo w erable by using the principles and a conditional v ersion of Shannon-McMillan- Breiman Theorem. The next three sections are de v oted to pro v e that a TD S is HUL iff it is asymptotically h -expansiv e, and the main ingredien ts of whic h are some tec hniques dev elop ed in [30, 5, 10, 19]. An example with finite entrop y whic h is hereditarily low erable but not HUL is presen ted at the end of pap er. W e thank D . F eng [12] for a sking the question: whether each non-empt y compact subset is lo w erable? His ques tion gav e us the first motiv a tion of the researc h. W e also thank the referees of the pap er fo r their careful reading a nd useful suggestions whic h greatly improv ed the writing of the pap er. 2. Preliminar y The discussions in this section and next section pro ceed f o r a gener al TDS (GTDS), b y a GTD S ( X , T ) w e mean a compact metric space X and a con tin uous mapping T : X → X . Let ( X, T ) be a G TDS, K ⊆ X and W a family o f subsets of X . Set diam( K ) to b e the diameter of K and put || W || = sup { diam( W ) : W ∈ W } . W e shall write K  W if K ⊆ W for some W ∈ W and else K  W . If W 1 is ano t her family of subsets of X , W is said to b e finer than W 1 (w e shall write W  W 1 ) when W  W 1 for eac h W ∈ W . W e shall say that a n umerical function incr e ases (resp. de cr e a s es ) with respect to (w.r.t.) a set v ariable K or a family v ariable W if t he v alue nev er decreases (resp. increases) when K is replaced b y a set K 1 with K 1 ⊆ K or when W is replaced b y a family W 1 with W 1  W . By a c ov e r of X w e mean a finite family of Borel subsets with union X and a p artition a cov er wh ose elemen ts are disjoin t. Denote by C X (resp. C o X , P X ) the set of cov ers (resp. op en co v ers, partitions). Observ e that if U ∈ C o X then U has a Lebesgue n um b er λ > 0 and so W  U when ||W || < λ . If α ∈ P X and x ∈ X then let α ( x ) b e the elemen t of α containing x . Giv en U 1 , U 2 ∈ C X , set U 1 ∨ U 2 = { U 1 ∩ U 2 : U 1 ∈ U 1 , U 2 ∈ U 2 } , ob viously U 1 ∨ U 2 ∈ C X and U 1 ∨ U 2  U 1 . U 1  U 2 need not imply that U 1 ∨ U 2 = U 1 , U 1  U 2 iff U 1 is equiv a lent to U 1 ∨ U 2 in the sense t ha t eac h refines the other. F or eac h U ∈ C X and a n y m, n ∈ Z + with m ≤ n w e set U n m = W n i = m T − i U . The following obvious fact will b e used in sev eral places and is easy t o chec k. Lemma 2.1. L et V ∈ C o X and {U n : n ∈ N } ⊆ C X . If ||U n || → 0 as n → + ∞ then ther e exists n 0 ∈ N such that U n  V for e ach n ≥ n 0 . 2.1. T op ological en trop y of subsets. 4 lo weri ng top olog ical entrop y ov er subsets Let ( X , T ) b e a GTDS, K ⊆ X and U ∈ C X . Set N ( U , K ) to b e the minimal cardinalit y of sub-families V ⊆ U with ∪V ⊇ K , where ∪V = S V ∈V V . W e write N ( U , ∅ ) = 0 b y con v en tion. Ob viously , N ( U , T ( K )) = N ( T − 1 U , K ). Let h U ( T , K ) = lim sup n → + ∞ 1 n log N ( U n − 1 0 , K ) . Clearly h U ( T , K ) increases w.r.t. U . Define the top olo gi c al entr opy of K by h ( T , K ) = sup U ∈C o X h U ( T , K ) , and define the top olo gic al entr opy of ( X, T ) b y h top ( T , X ) = h ( T , X ). Let Z be a top ological metric space a nd f : Z → [ −∞ , + ∞ ] a generalized real- v alued function on Z . The function f is called upp er semi-c ontinuous (u.s.c. for short) if { z ∈ Z : f ( z ) ≥ r } is a closed subset of Z for eac h r ∈ R , equiv alen tly , lim sup z ′ → z f ( z ′ ) ≤ f ( z ) for eac h z ∈ Z . Th us, the infimum of any family of u.s.c. functions is a gain a u.s.c. one, b oth the sum and suprem um of finitely man y u.s.c. functions are u.s.c. ones. In pa rticular, the infim um of any family o f contin uous functions is a u.s.c. f unction. Let ( X , T ) b e a GTDS and 2 X its h yp erspace, that is, 2 X = { K : K is a non-empt y compact subset of X } . W e endo w the Hausdorff metric on 2 X . Then T induces a con tin uous mapping b T o n 2 X b y b T ( K ) = T K . The entr opy hyp er-function H : 2 X → [0 , h top ( T , X )] of ( X , T ) is defined b y H ( K ) = h ( T , K ) for K ∈ 2 X . Then w e hav e the follow results. Prop osition 2.2. L et ( X , T ) b e a GTDS and U ∈ C o X . Then (1) h U ( T , K ) = h U ( T , T K ) for any K ⊆ X . Mor e over, the entr opy hyp er- function H is b T -inva riant. (2) The function h U ( T , • ) is Bor el me asur a ble on 2 X . (3) The entr opy h yp er-function H is Bor el me asur a ble. Pr o of. (1) is clear. (2) follows fr om the f ollo wing fact that for an y V ∈ C o X , N ( V , • ) : K ∈ 2 X 7→ N ( V , K ) is a u.s.c function on 2 X . ( 3 ) comes fro m (2).  W e ma y also obtain the top ological en trop y of su bsets using Bo w en’s separated and spanning sets (see [29, P 168 − 174 ]). L et ( X , T ) b e a TDS with d a metric on X . F or eac h n ∈ N we define a new metric d n on X by d n ( x, y ) = max 0 ≤ i ≤ n − 1 d ( T i x, T i y ) . Let ǫ > 0 and K ⊆ X . A subset F of X is said to ( n, ǫ ) -s p an K w.r.t. T if for eac h x ∈ K , there is y ∈ F with d n ( x, y ) ≤ ǫ ; a subset E o f K is said to b e ( n, ǫ ) - sep a r ate d w.r.t. T if x, y ∈ E , x 6 = y implies d n ( x, y ) > ǫ . Let r n ( d, T , ǫ, K ) denote the smallest cardinality of an y ( n, ǫ )-spanning set fo r K w.r.t. T and s n ( d, T , ǫ, K ) W en Hu ang, Xiangdong Y e and Guoh ua Zhang 5 denote the largest cardinalit y of any ( n, ǫ )- separated subset of K w.r.t. T . W e write r n ( d, T , ǫ, ∅ ) = s n ( d, T , ǫ, ∅ ) = 0 b y con v en tion. Put r ( d, T , ǫ, K ) = lim sup n → + ∞ 1 n log r n ( d, T , ǫ, K ) and s ( d, T , ǫ, K ) = lim sup n → + ∞ 1 n log s n ( d, T , ǫ, K ) . Then put h ∗ ( d, T , K ) = lim ǫ → 0+ r ( d, T , ǫ, K ) a nd h ∗ ( d, T , K ) = lim ǫ → 0+ s ( d, T , ǫ, K ) . It is w ell known that h ∗ ( d, T , K ) = h ∗ ( d, T , K ) is indep enden t of the choice of a compatible metric d on the space X . Now , if U ∈ C o X has a Leb esgue n um b er δ > 0 then, for an y δ ′ ∈ (0 , δ 2 ) a nd each V ∈ C o X with ||V || ≤ δ ′ , one has N ( U n − 1 0 , K ) ≤ r n ( d, T , δ ′ , K ) ≤ s n ( d, T , δ ′ , K ) ≤ N ( V n − 1 0 , K ) for eac h n ∈ N . So if {U n } n ∈ N ⊆ C o X satisfies ||U n || → 0 as n → + ∞ then h ∗ ( d, T , K ) = h ∗ ( d, T , K ) = lim n → + ∞ h U n ( T , K ) = h ( T , K ) . It is also ob vious that h ( T , K ) = h ( T , K ). 2.2. Dimensional en tropy of subsets. In the pro cess of prov ing that eac h TD S with finite entrop y is low erable, w e shall use some concept named dimensional en trop y of subse ts, which is another kind of top ological en trop y in tro duced a nd studied in [4]. Let’s see how to define it. Let ( X , T ) b e a GTDS and U ∈ C X . F o r K ⊆ X let n T , U ( K ) =    0 , if K  U ; + ∞ , if T i K  U for all i ∈ Z + ; k , k = max { j ∈ N : T i ( K )  U for each 0 ≤ i ≤ j − 1 } . F or k ∈ N , we define C ( T , U , K, k ) = {E : E is a countable family of subsets of X suc h that K ⊆ ∪E and E  U k − 1 0 } . Then for eac h λ ∈ R set m T , U ( K , λ, k ) = inf E ∈ C ( T , U ,K ,k ) m ( T , U , E , λ ) , where m ( T , U , E , λ ) = P E ∈E e − λn T , U ( E ) and w e write m ( T , U , ∅ , λ ) = 0 b y conv en- tion. As m T , U ( K , λ, k ) is decreasing w.r.t. k , we can define m T , U ( K , λ ) = lim k → + ∞ m T , U ( K , λ, k ) . Notice that m T , U ( K , λ ) ≤ m T , U ( K , λ ′ ) fo r λ ≥ λ ′ and m T , U ( K , λ ) / ∈ { 0 , + ∞ } for at most one λ [4]. W e define the d i m ensional entr opy o f K r elative to U b y h B U ( T , K ) = inf { λ ∈ R : m T , U ( K , λ ) = 0 } = sup { λ ∈ R : m T , U ( K , λ ) = + ∞ } . 6 lo weri ng top olog ical entrop y ov er subsets The d imensional entr opy of K is defined by h B ( T , K ) = sup U ∈C o X h B U ( T , K ) . Note that h B U ( T , K ) increases w.r.t. U ∈ C X , th us if {U n } n ∈ N ⊆ C o X satisfies lim n → + ∞ ||U n || = 0 then lim n → + ∞ h B U n ( T , K ) = h B ( T , K ). The following results ar e elemen tary (see for example [4, Prop ositions 1 and 2]). Prop osition 2.3. L et ( X , T ) b e a GTDS, K 1 , K 2 , · · · , K ⊆ X and U ∈ C X . Then (1) h U ( T , X ) = h B U ( T , X ) if U ∈ C o X , so h ( T , X ) = h B ( T , X ) . (2) h B U ( T , S n ∈ N K n ) = sup n ∈ N h B U ( T , K n ) , so h B ( T , [ n ∈ N K n ) = sup n ∈ N h B ( T , K n ) . (3) F or e ach m ∈ N and i ≥ 0 , h B T − i U ( T m , K ) ≥ h B U ( T m , T i K ) , so h B ( T m , K ) ≥ h B ( T m , T i K ) . (4) F or e a c h m ∈ N , h B U m − 1 0 ( T m , K ) = mh B U ( T , K ) , s o h B ( T m , K ) = mh B ( T , K ) . Pr o of. (1) is [4, Prop osition 1 ]. (2) is obvious . (3) Let m ∈ N and i ≥ 0. Assume k ∈ N and λ > 0. If E ∈ C ( T m , T − i U , K , k ) then n T m , U ( T i E ) = n T m ,T − i U ( E ) ≥ k fo r eac h E ∈ E and so T i ( E ) . = { T i E : E ∈ E } ∈ C ( T m , U , T i K, k ) , th us m T m , U ( T i K, λ, k ) ≤ m ( T m , U , T i ( E ) , λ ) = X E ∈E e − λn T m , U ( T i E ) = X E ∈E e − λn T m ,T − i U ( E ) = m ( T m , T − i U , E , λ ) , whic h implies m T m , U ( T i K, λ, k ) ≤ m T m ,T − i U ( K , λ, k ) as E is arbitrary . Letting k → + ∞ w e get m T m , U ( T i K, λ ) ≤ m T m ,T − i U ( K , λ ), hence h B U ( T m , T i K ) ≤ h B T − i U ( T m , K ), as λ > 0 is a rbitrary . (4) Let m ∈ N and n ∈ N , λ > 0. If E ∈ C ( T , U , K , mn ) t hen n T m , U m − 1 0 ( E ) =  n T , U ( E ) m  ≥ max  n, n T , U ( E ) m − m − 1 m  for eac h E ∈ E , where [ a ] denotes the in tegral part of a real num b er a , so inf E ∈E n T m , U m − 1 0 ( E ) ≥ n, th us E ∈ C ( T m , U m − 1 0 , K , n ) and m T m , U m − 1 0 ( K , λ, n ) ≤ m ( T m , U m − 1 0 , E , λ ) = X E ∈E ( e λ ) − n T m , U m − 1 0 ( E ) ≤ X E ∈E ( e λ ) m − 1 m − n T , U ( E ) m = e ( m − 1) λ m · m ( T , U , E , λ m ) , W en Hu ang, Xiangdong Y e and Guoh ua Zhang 7 whic h implies m T m , U m − 1 0 ( K , λ, n ) ≤ e ( m − 1) λ m m T , U ( K , λ m , mn ) as E is arbitrary . W e get m T m , U m − 1 0 ( K , λ ) ≤ e ( m − 1) λ m · m T , U ( K , λ m ) b y letting n → + ∞ , hence h B U m − 1 0 ( T m , K ) ≤ mh B U ( T , K ), as λ > 0 is arbitrary . F ollowing similar discussions w e obtain m T , U ( K , λ ) ≤ m T m , U m − 1 0 ( K , mλ ) fo r eac h λ > 0, then h B U ( T , K ) ≤ 1 m h B U m − 1 0 ( T m , K ). That is, h B U m − 1 0 ( T m , K ) = mh B U ( T , K ).  By Prop osition 2.3 (2), h B ( T , E ) increases w.r.t. E ⊆ X . A t the same time, if E ⊆ X is a non-empt y countable set then h B ( T , E ) = 0. Finally , it is w orth men tioning that a). h B U ( T , ∅ ) = h U ( T , ∅ ) = −∞ for any U ∈ C X , and so h B ( T , ∅ ) = h ( T , ∅ ) = −∞ ; b). when ∅ 6 = K ⊆ X , one ha s h U ( T , K ) ≥ h B U ( T , K ) ≥ 0 for an y U ∈ C X , a nd so h ( T , K ) ≥ h B ( T , K ) ≥ 0. 3. Dist ribution princip les In this section w e shall presen t tw o imp ortan t distribution principles whic h link Question 1.2 with ergo dic theory and pla y a k ey role in the next section. W e remark that the distribution principles w ere essen tially contained in [26]. The first result is an obvious link b et w een tw o definitions of en trop y . Lemma 3.1 ( Br idge Lemma). L et ( X , T ) b e a GTDS, U ∈ C X and K ⊆ X . Th en h B U ( T , K ) ≤ lim inf n → + ∞ 1 n log N ( U n − 1 0 , K ) ≤ h U ( T , K ) . Pr o of. When K = ∅ , this is clear. Now w e assume K 6 = ∅ . F or each n ∈ N let T n = { A 1 , · · · , A N ( U n − 1 0 ,K ) } ⊆ U n − 1 0 suc h that ∪T n ⊇ K . As n T , U ( A ) ≥ n for eac h A ∈ T n , for eac h λ ≥ 0 one has m T , U ( K , λ, n ) ≤ X A ∈T n  e λ  − n T , U ( A ) ≤ X A ∈T n ( e λ ) − n = N ( U n − 1 0 , K ) e − λn , then m T , U ( K , λ ) ≤ lim inf n → + ∞ N ( U n − 1 0 , K ) e − λn = lim inf n → + ∞ e − n ( λ − 1 n log N ( U n − 1 0 ,K )) . So, if λ > lim inf n → + ∞ 1 n log N ( U n − 1 0 , K ) then m T , U ( K , λ ) = 0 , whic h ends the pro of.  Let ( X , T ) b e a G TDS, U ∈ C X , K ⊆ X and n ∈ N . Set M ( T , U , K , n ) to b e the collection of all countable fa milies T of subsets of X with ∪T ⊇ K and fo r eac h A ∈ T , A ∩ K 6 = ∅ , n T , U ( A ) ≥ n and A ∈ U n T , U ( A ) − 1 0 . Then for eac h λ ∈ R set f T , U ( K , λ ) = lim n → + ∞ inf T ∈ M ( T , U ,K,n ) m ( T , U , T , λ ) . It’s not hard to ch ec k that f T , U ( K , λ ) = m T , U ( K , λ ) for λ ∈ R . In f act, f o r E ∈ C ( T , U , K, n ), note that for eac h E ∈ E there exists e E ∈ U n T , U ( E ) − 1 0 with E ⊆ 8 lo weri ng top olog ical entrop y ov er subsets e E and so n T , U ( E ) = n T , U ( e E ). Then let T . = { e E : E ∈ E with E ∩ K 6 = ∅} . Then T ∈ M ( T , U , K , n ). Particularly , when E ∩ K 6 = ∅ for each E ∈ E , one has m ( T , U , T , λ ) = m ( T , U , E , λ ). This implies f T , U ( K , λ ) = m T , U ( K , λ ). F or a GTDS ( X , T ), denote b y M ( X ) t he set of all Borel probability measures on X . The following tw o principles will b e prov ed to b e v ery useful. Lemma 3.2 ( N on-Uni form Mass Distribution Principle). L et ( X , T ) b e a GTDS, d > 0 , M ∈ N , Z ⊆ X , α ∈ P X , U ∈ C X and θ ∈ M ( X ) . Assume that e ach element of U has a non-empty interse ction with at most M eleme n ts of α . If ther e exists Z θ ⊆ Z such that Z θ has p o sitive outer θ -me asur e (i.e. θ ∗ ( Z θ ) > 0 ) and ∀ x ∈ Z θ , ∃ c ( x ) > 0 such that ∀ n ∈ N , θ ( α n − 1 0 ( x )) ≤ c ( x ) e − nd . Then h B U ( T , Z ) ≥ d − log M . In p articular, h B α ( T , Z ) ≥ d . Pr o of. It mak es no difference to assume d − log M > 0 . F or each k ∈ N set Z k θ = { x ∈ Z θ : c ( x ) ≤ k } . Then for some N ∈ N , θ ∗ ( Z N θ ) > 0, as Z 1 θ ⊆ Z 2 θ ⊆ · · · , Z θ = S k ∈ N Z k θ and θ ∗ ( Z θ ) > 0. Let n ∈ N and T ∈ M ( T , U , Z , n ). If A ∈ T satisfies A ∩ Z N θ 6 = ∅ , then for eac h s ∈ N and B ∈ α min { n T , U ( A ) ,s }− 1 0 with B ∩ ( A ∩ Z N θ ) 6 = ∅ select x B ∈ B ∩ ( A ∩ Z N θ ), so θ ( B ) = θ ( α min { n T , U ( A ) ,s }− 1 0 ( x B )) ≤ c ( x B ) e − min { n T , U ( A ) ,s } d ≤ N e − min { n T , U ( A ) ,s } d . Since there are at most M min { n T , U ( A ) ,s } elemen ts of α min { n T , U ( A ) ,s }− 1 0 whic h ha v e no n- empt y inte rsection with A ∩ Z N θ , we hav e θ ∗ ( A ∩ Z N θ ) ≤ M min { n T , U ( A ) ,s } N e − min { n T , U ( A ) ,s } d = N e − min { n T , U ( A ) ,s } ( d − log M ) . Letting s → + ∞ w e obtain that θ ∗ ( A ∩ Z N θ ) ≤ N e − n T , U ( A )( d − log M ) for an y A ∈ T satisfying A ∩ Z N θ 6 = ∅ . Moreo v er, X A ∈T e − n T , U ( A )( d − log M ) ≥ X A ∈T ,A ∩ Z N θ 6 = ∅ e − n T , U ( A )( d − log M ) ≥ X A ∈T ,A ∩ Z N θ 6 = ∅ θ ∗ ( A ∩ Z N θ ) N ≥ 1 N θ ∗ ( Z N θ ) > 0 . Since n and T ar e ar bit r a ry , f T , U ( Z , d − log M ) ≥ 1 N θ ∗ ( Z N θ ) > 0. So h B U ( T , Z ) ≥ d − log M .  Lemma 3.3 ( Uniform Mass Distribution Principle). L et ( X, T ) b e a GTDS, c > 0 , d > 0 , Z ⊆ X and α ∈ P X . If ther e exists θ ∈ M ( X ) such that for e ac h x ∈ Z and n ∈ N , θ ( α n − 1 0 ( x )) ≥ ce − nd . T hen h α ( T , Z ) ≤ d . Pr o of. If Z = ∅ then h α ( T , Z ) = −∞ ≤ d . In the f o llo wing w e assume Z 6 = ∅ . F or n ∈ N , let T n = { A 1 , · · · , A k } b e the collection of all elemen ts of α n − 1 0 whic h ha v e non-empt y intersec tion with Z , where k = N ( α n − 1 0 , Z ). T ake x i ∈ A i ∩ Z , then θ ( A i ) = θ ( α n − 1 0 ( x i )) ≥ ce − nd for i ∈ { 1 , · · · , k } . Therefore 1 ≥ P k i =1 θ ( A i ) ≥ k ce − nd , that is, N ( α n − 1 0 , Z ) = k ≤ e nd c . Finally letting n → + ∞ w e kno w h α ( T , Z ) ≤ d .  W en Hu ang, Xiangdong Y e and Guoh ua Zhang 9 4. Each TDS with finite entr opy is lowerable In this section w e shall give an affirmativ e answ er to Question 1.2 for a TD S with finite en trop y . In fact, we can obtain mor e ab out it. Precisely , if ( X , T ) is a TDS with finite entrop y , then for each 0 ≤ h ≤ h top ( T , X ) there exists a non- empt y compact subset K h ⊆ X suc h that h B ( T , K h ) = h ( T , K h ) = h (for details see Theorem 4.4), particularly , ( X, T ) is lo w erable. Let ( X , T ) b e a GTDS. Denote b y M ( X , T ) and M e ( X , T ) the set of all T - in v ariant Borel probabilit y measures and ergo dic T -in v a rian t Borel probabilit y mea- sures on X , resp ectiv ely . Then M ( X ) a nd M ( X , T ) are b oth con vex , compact metric spaces when endo w ed with the we ak ∗ -top ology . Denote b y B X the set of all Borel subsets of X . F or an y giv en α ∈ P X , µ ∈ M ( X ) and any sub- σ -alg ebra C ⊆ B µ , where B µ is the completion of B X under µ , the c o nditional informational function of α r elevant to C is defined b y I µ ( α |C )( x ) = X A ∈ α − 1 A ( x ) log E µ (1 A |C )( x ) , where E µ (1 A |C ) is the conditional exp ectation of 1 A w.r.t. C . Let H µ ( α |C ) = Z X I µ ( α |C )( x ) dµ ( x ) = X A ∈ α Z X − E µ (1 A |C ) log E µ (1 A |C ) dµ. A standard fact states that H µ ( α |C ) increases w.r.t. α and decreases w.r.t. C . No w set H µ ( U |C ) = inf β ∈P X ,β U H µ ( β |C ) for U ∈ C X . Clearly , H µ ( U |C ) increases w.r.t. U and decreases w.r.t. C . When µ ∈ M ( X , T ) and T − 1 C ⊆ C in the sense of µ , it is not hard to see that H µ ( U n − 1 0 |C ) is a no n- negativ e and sub-additiv e sequence for a giv en U ∈ C X , so w e can define h µ ( T , U |C ) = lim n → + ∞ 1 n H µ ( U n − 1 0 |C ) = inf n ≥ 1 1 n H µ ( U n − 1 0 |C ) . Clearly , h µ ( T , U |C ) also increases w.r.t. U and decreases w.r.t. C . The r elative me a s ur e-the or etic al µ -entr opy of ( X , T ) r elevant to C is defined b y h µ ( T , X |C ) = sup α ∈P X h µ ( T , α |C ) . F ollowing a similar discussion of [1 5, Lemma 2.3 (1)], one has h µ ( T , X |C ) = sup U ∈C o X h µ ( T , U |C ) . So, if { U n } n ∈ N ⊆ C o X satisfies lim n → + ∞ ||U n || = 0 then lim n → + ∞ h µ ( T , U n |C ) = h µ ( T , X |C ), as h µ ( T , U |C ) increases w.r.t. U . It is not hard to see that h µ ( T , U |C ) = 1 n h µ ( T n , U n − 1 0 |C ) for eac h n ∈ N and U ∈ C X . 10 lo weri ng top olog ical entrop y ov er subsets If C = {∅ , X } (mo d µ ), for simplicit y we shall write H µ ( U |C ), h µ ( T , U |C ) and h µ ( T , X |C ) b y H µ ( U ), h µ ( T , U ) and h µ ( T , X ), r esp ectiv ely . The following result is a conditional v ersion of Shannon-McMillan-Breiman The- orem. Its pro of is completely similar t o the pro o f of Shannon- McMillan-Breiman Theorem (see e.g. [1, Theorem 4.2 ], [17, Theorem I I.1.5], [14 ]) . Theorem 4.1. L et ( X, T ) b e a TDS, µ ∈ M ( X , T ) , α ∈ P X and C ⊆ B µ a T - invariant sub- σ -algebr a (i.e. T − 1 C = C in the sense of µ ) . The n ther e exists a T -inva riant function f ∈ L 1 ( µ ) such that R X f ( x ) dµ ( x ) = h µ ( T , α |C ) and lim n → + ∞ I µ ( α n − 1 0 |C )( x ) n = f ( x ) for µ -a.e. x ∈ X and in L 1 ( µ ) . Mor e over, if µ is er go di c then f ( x ) = h µ ( T , α |C ) for µ -a.e. x ∈ X . Let ( X , T ) be a TDS, µ ∈ M ( X , T ) and B µ the completion of B X under µ . Then ( X, B µ , µ, T ) is a Leb esgue system. If { α i } i ∈ I is a coun table family of finite partitions of X , the partition α = W i ∈ I α i is called a me asur able p artition . The sets A ∈ B µ , whic h are unions of atoms of α , form a sub- σ -algebra of B µ denoted by b α or α if t here is no am biguit y . Ev ery sub- σ -a lgebra of B µ coincides with a sub- σ -algebra constructed in this w a y (mo d µ ) . Giv en a measurable partit io n α , put α − = W + ∞ n =1 T − n α and α T = W + ∞ n = −∞ T − n α . Define in the same w ay C − and C T if C is a sub- σ -a lgebra of B µ . Clearly , for a measurable pa r tition α , c α − = ( b α ) − (mo d µ ) and c α T = ( b α ) T (mo d µ ) . Let C b e a sub- σ - algebra of B µ and α the measurable partition of X with b α = C (mo d µ ). µ can b e disin tegrated o v er C as µ = R X µ x dµ ( x ) where µ x ∈ M ( X ) and µ x ( α ( x )) = 1 fo r µ -a .e. x ∈ X . The disin tegration is c haracterized b y (4 .1 ) and (4.2) b elo w: for ev ery f ∈ L 1 ( X , B X , µ ), f ∈ L 1 ( X , B X , µ x ) f or µ -a .e. x ∈ X and (4.1) the function x 7→ Z X f ( y ) dµ x ( y ) is in L 1 ( X , C , µ ); (4.2) E µ ( f |C )( x ) = Z X f dµ x for µ -a.e. x ∈ X . Then, for any f ∈ L 1 ( X , B X , µ ), the following holds Z X  Z X f dµ x  dµ ( x ) = Z X f dµ. Define fo r µ -a.e. x ∈ X the set Γ x = { y ∈ X : µ x = µ y } . Then µ x (Γ x ) = 1 for µ -a.e. x ∈ X . Hence giv en any f ∈ L 1 ( X , B X , µ ), for µ - a .e. x ∈ X , one has E µ ( f |C )( y ) = Z X f dµ y = Z X f dµ x = E µ ( f |C )( x ) (4.3) for µ x -a.e. y ∈ X . P articularly , if f is C -measurable, then f or µ -a.e. x ∈ X , one has f ( y ) = f ( x ) f o r µ x -a.e. y ∈ X . (4.4) W en Hu ang, Xiangdong Y e and Guoh ua Zhang 11 Prop osition 4.2. L et ( X, T ) b e a TDS, µ ∈ M ( X , T ) and C ⊆ B µ a T -invariant sub- σ -algebr a. I f µ = R X µ x dµ ( x ) is the disinte gr ation of µ over C , then (1) L e t U ∈ C X and α ∈ P X such that e ach element of U has a non-em pty interse ction with at most M elements of α ( M ∈ N ). If f C T ,α ( x ) is the function obtaine d in The or em 4.1 for T , α and C , then for µ -a.e . x ∈ X , h B U ( T , Z x ) ≥ f C T ,α ( x ) − log M for any Z x ∈ B X with µ x ( Z x ) > 0 . Particularly, if µ is er go dic, then for µ -a.e. x ∈ X , h B U ( T , Z x ) ≥ h µ ( T , α |C ) − log M for any Z x ∈ B X with µ x ( Z x ) > 0 . (2) If µ is er go dic then, for µ -a.e. x ∈ X , when Z x ∈ B X with µ x ( Z x ) > 0 one has h B ( T , Z x ) ≥ h µ ( T , X |C ) . Pr o of. (1) Note t hat lim n → + ∞ I µ ( α n − 1 0 |C )( x ) n = f C T ,α ( x ) for µ - a.e. x ∈ X and f C T ,α is C - measurable, using (4.3) for all 1 B , B ∈ α n − 1 0 and (4.4) for f C T ,α , there exis ts X ∞ ∈ B X with µ ( X ∞ ) = 1 suc h that for eac h x ∈ X ∞ , one can find W x ∈ B X with µ x ( W x ) = 1 and if y ∈ W x then (a). lim n → + ∞ I µ ( α n − 1 0 |C )( y ) n = f C T ,α ( y ) = f C T ,α ( x ). (b). E µ (1 B |C )( y ) = E µ (1 B |C )( x ) = µ x ( B ) for an y B ∈ α n − 1 0 and eac h n ∈ N . Moreo v er, fo r a ny y ∈ W x , where x ∈ X ∞ , one has lim n → + ∞ − log µ x ( α n − 1 0 ( y )) n = lim n → + ∞ − log E µ (1 α n − 1 0 ( y ) |C )( y ) n (4.5) = lim n → + ∞ I µ ( α n − 1 0 |C )( y ) n = f C T ,α ( x ) . F or a given x ∈ X ∞ let Z x ∈ B X with µ x ( Z x ) > 0. Clearly µ x ( Z x ∩ W x ) = µ x ( Z x ) > 0. F or δ > 0 and ℓ ∈ N , w e define Z ℓ x ( δ ) = { y ∈ Z x ∩ W x : µ x ( α n − 1 0 ( y )) ≤ e − n ( f C T ,α ( x ) − δ ) for eac h n ≥ ℓ } . Then S + ∞ ℓ =1 Z ℓ x ( δ ) = Z x ∩ W x b y (4 .5). Hence there exists N ∈ N suc h t ha t µ x ( Z N x ( δ )) > 0. F or y ∈ Z N x ( δ ), as µ x ( α n − 1 0 ( y )) ≤ e − n ( f C T ,α ( x ) − δ ) for each n ≥ N , one has µ x ( α n − 1 0 ( y )) ≤ c ( y ) e − n ( f C T ,α ( x ) − δ ) for an y n ∈ N , where c ( y ) = max { 1 , P N − 1 i =1 e i ( f C T ,α ( x ) − δ ) } ∈ (0 , + ∞ ). Th us applying Lemma 3.2 to Z N x ( δ ) we obtain h B U ( T , Z N x ( δ )) ≥ f C T ,α ( x ) − δ − log M , so h B U ( T , Z x ) ≥ f C T ,α ( x ) − δ − log M . Note t ha t the last inequality is true for any δ > 0, one has h B U ( T , Z x ) ≥ f C T ,α ( x ) − log M . (2) F or k ∈ N we tak e U k ∈ C o X with | | U k || ≤ 1 k . Using [4, L emma 2] f o r eac h n ∈ N there exists α n,k ∈ P X suc h that α n,k  ( U k ) n − 1 0 and at most n # U k elemen ts 12 lo weri ng top olog ical entrop y ov er subsets of α n,k can ha v e a p oin t in all their closures, here # U k means the cardinality of U k . It’s easy to construct U n,k ∈ C o X suc h that eac h elemen t of U n,k has a no n-empt y in tersection with at most n # U k elemen ts of α n,k (see also [4, Lemma 1]). No w assume that µ is ergo dic. Let f C T n ,α n,k ( x ) b e the function obtained in Theorem 4.1 for T n , α n,k and C . Then f C T n ,α n,k ( x ) is T n -in v a r ia n t and R X f C T n ,α n,k ( x ) dµ ( x ) = h µ ( T n , α n,k |C ). Let g k n ( x ) = 1 n P n − 1 i =0 f C T n ,α n,k ( T i x ). Then g k n ( x ) is T -in v aria nt, as f C T n ,α n,k ( x ) is T n -in v a r ia n t. Moreo v er, since µ ∈ M e ( X , T ), g k n ( x ) is constan t and g k n ( x ) ≡ Z X g k n ( y ) dµ ( y ) = 1 n n − 1 X i =0 Z X f C T n ,α n,k ( T i y ) dµ ( y ) = 1 n n − 1 X i =0 Z X f C T n ,α n,k ( y ) dµ ( y ) = h µ ( T n , α n,k |C ) (4.6) for µ -a.e. x ∈ X . By (1) for µ - a .e. x ∈ X , if Z x ∈ B X with µ x ( Z x ) > 0 then (4.7) h B U n,k ( T n , Z x ) ≥ f C T n ,α n,k ( x ) − log ( n # U k ) for eac h k ∈ N , n ∈ N . Moreo v er, note that T µ x = µ T x for µ -a.e. x ∈ X , there exists a T -inv arian t subset X 1 ⊆ X with µ ( X 1 ) = 1 suc h that T µ x = µ T x and b oth (4.6) and (4.7) hold for all x ∈ X 1 . No w for any given x ∈ X 1 let Z x ∈ B X with µ x ( Z x ) > 0. Then T i x ∈ X 1 and µ T i x ( T i Z x ) = T i µ x ( T i Z x ) = µ x ( Z x ) > 0 for an y i ≥ 0 . By (4.7), for eac h k ∈ N , n ∈ N and i ≥ 0 , h B ( T n , T i Z x ) ≥ h B U n,k ( T n , T i Z x ) ≥ f C T n ,α n,k ( T i x ) − log ( n # U k ) . F or eac h n ∈ N , a s h B ( T n , Z x ) ≥ h B ( T n , T i Z x ) for eac h i ≥ 0 (see Prop osition 2 .3 (3)), w e hav e h B ( T n , Z x ) ≥ 1 n n − 1 X i =0 h B ( T n , T i Z x ) ≥ 1 n n − 1 X i =0  f C T n ,α n,k ( T i x ) − log ( n # U k )  = g k n ( x ) − log ( n # U k ) = h µ ( T n , α n,k |C ) − log ( n # U k ) for all k ∈ N . Then using Prop osition 2.3 (4) w e ha v e h B ( T , Z x ) = h B ( T n , Z x ) n ≥ 1 n ( h µ ( T n , α n,k |C ) − log( n # U k )) ≥ 1 n ( h µ ( T n , ( U k ) n − 1 0 |C ) − log ( n # U k )) = h µ ( T , U k |C ) − log( n # U k ) n for eac h k ∈ N a nd n ∈ N . No w fixing k ∈ N letting n → + ∞ w e get h B ( T , Z x ) ≥ h µ ( T , U k |C ). Finally letting k → + ∞ w e hav e h B ( T , Z x ) ≥ lim k → + ∞ h µ ( T , U k |C ) = h µ ( T , X |C ). This completes the pro o f o f (2) since µ ( X 1 ) = 1.  The following result is an application of Prop osition 4.2. Lemma 4.3. L e t ( X, T ) b e a TDS, µ ∈ M e ( X , T ) and C ⊆ B µ a T -inv a riant sub- σ -algebr a. I f µ = R X µ x dµ ( x ) is the disinte gr ation of µ over C , then W en Hu ang, Xiangdong Y e and Guoh ua Zhang 13 (1) If α ∈ P X then for µ -a. e . x ∈ X , fixi n g e ach x , for e ach ǫ ∈ (0 , 1) ther e exists a c o m p act subset Z x ( α, ǫ ) of X such that µ x ( Z x ( α, ǫ )) ≥ 1 − ǫ an d h B α ( T , Z x ( α, ǫ )) = h α ( T , Z x ( α, ǫ )) = h µ ( T , α |C ) . (2) F or µ -a.e. x ∈ X , fixing e ach x , for e ach ǫ ∈ (0 , 1) ther e exists a c omp act subset Z x ( ǫ ) of X such that µ x ( Z x ( ǫ )) ≥ 1 − ǫ and h B ( T , Z x ( ǫ )) = h ( T , Z x ( ǫ )) = h µ ( T , X |C ) . Pr o of. (1) Let α ∈ P X . As µ ∈ M e ( X , T ), b y (4.5) there exists X ∞ ∈ B X with µ ( X ∞ ) = 1 suc h that for eac h x ∈ X ∞ , o ne can find W x ∈ B X with µ x ( W x ) = 1 suc h that for eac h y ∈ W x lim n → + ∞ − 1 n log µ x ( α n − 1 0 ( y )) = h µ ( T , α |C ) (for details see the pro of of Prop osition 4.2 (1)). By Prop osition 4.2 ( 1 ), w.l.g. we ma y require (4.8) h B α ( T , Z ) ≥ h µ ( T , α |C ) for an y x ∈ X ∞ and Z ∈ B X with µ x ( Z ) > 0 (if necessary w e take a subset of X ∞ ). Let x ∈ X ∞ . Ob viously , for each y ∈ W x and m ∈ N there exists n y , m ∈ N suc h that if n ≥ n y , m then − 1 n log µ x ( α n − 1 0 ( y )) ≤ h µ ( T , α |C ) + 1 m , i.e. µ x ( α n − 1 0 ( y )) ≥ e − n ( h µ ( T ,α |C )+ 1 m ) . So fo r each n ∈ N , µ x ( α n − 1 0 ( y )) ≥ e − ( n + n y ,m )( h µ ( T ,α |C )+ 1 m ) . W e in tro duce a µ x -measurable function by defining for µ x -a.e. y ∈ X c m ( y ) = inf n ∈ N µ x ( α n − 1 0 ( y )) e − n ( h µ ( T ,α |C )+ 1 m ) ≥ e − n y ,m ( h µ ( T ,α |C )+ 1 m ) > 0 . Let Z m k = { y ∈ W x : c m ( y ) ≥ 1 k } f or eac h k ∈ N . Then Z m k is µ x -measurable and h α ( T , Z m k ) ≤ h µ ( T , α |C ) + 1 m b y Lemma 3 .3. Moreo v er, a s lim k → + ∞ µ x ( Z m k ) = 1, for eac h ǫ ∈ (0 , 1) there exists a compact subset B m ǫ ⊆ Z m K for some K ∈ N such that µ x ( X \ B m ǫ ) < ǫ 2 m and h α ( T , B m ǫ ) ≤ h µ ( T , α |C ) + 1 m . F or ǫ ∈ (0 , 1) , set Z x ( α, ǫ ) = T m ∈ N B m ǫ . Then Z x ( α, ǫ ) is a compact subset of X , µ x ( Z x ( α, ǫ )) = 1 − µ [ m ∈ N X \ B m ǫ ! ≥ 1 − X m ∈ N µ ( X \ B m ǫ ) ≥ 1 − ǫ > 0 and h α ( T , Z x ( α, ǫ )) ≤ inf m ∈ N h α ( T , B m ǫ ) ≤ inf m ∈ N  h µ ( T , α |C ) + 1 m  = h µ ( T , α |C ) . Moreo v er, using Lemma 3.1 and (4.8) w e hav e h B α ( T , Z x ( α, ǫ )) = h α ( T , Z x ( α, ǫ )) = h µ ( T , α |C ) . (2) Let {U n } n ∈ N ⊆ C o X with lim n → + ∞ ||U n || = 0. F or n ∈ N w e tak e α n ∈ P X with α n  U n . By (1) t here exists a measurable subset X ′ of X with µ ( X ′ ) = 1 suc h that 14 lo weri ng top olog ical entrop y ov er subsets if x ∈ X ′ then for each ǫ ∈ (0 , 1) and n ∈ N there exists a compact subset Z x ( n, ǫ ) suc h that µ x ( Z x ( n, ǫ )) ≥ 1 − ǫ 2 n and h U n ( T , Z x ( n, ǫ )) ≤ h α n ( T , Z x ( n, ǫ )) = h µ ( T , α n |C ) . By Prop osition 4.2 (2), w.l.g. (if necess ary w e take a subset o f X ′ ) we ma y require (4.9) h B ( T , Z ) ≥ h µ ( T , X |C ) for an y x ∈ X ′ and Z ∈ B X with µ x ( Z ) > 0. Let x ∈ X ′ . F or ǫ ∈ (0 , 1), Set Z x ( ǫ ) = T n ∈ N Z x ( n, ǫ ). Then Z x ( ǫ ) is a compact subset of X , µ x ( Z x ( ǫ )) = 1 − µ [ n ∈ N X \ Z x ( n, ǫ ) ! ≥ 1 − X n ∈ N µ ( X \ Z x ( n, ǫ )) ≥ 1 − ǫ > 0 and h ( T , Z x ( ǫ )) = sup n ∈ N h U n ( T , Z x ( ǫ )) ≤ sup n ∈ N h U n ( T , Z x ( n, ǫ )) ≤ sup n ∈ N h µ ( T , α n |C ) ≤ h µ ( T , X |C ) . Moreo v er, using Lemma 3 .1 and (4 .9) w e ha v e if x ∈ X ′ then h B ( T , Z x ( ǫ )) = h ( T , Z x ( ǫ )) = h µ ( T , X |C ). This finishes the pro of o f (2 ) since µ ( X ′ ) = 1.  With the ab o v e preparations w e can obtain the main result of this section. Theorem 4.4. L et ( X , T ) b e a TD S with fini te entr opy. Then for e ach 0 ≤ h ≤ h top ( T , X ) ther e exists a non -empty c omp act subset K h of X such that h B ( T , K h ) = h ( T , K h ) = h . In p articular, ( X , T ) is lowe r able. Pr o of. If h = h top ( T , X ), it is true for K h = X b y Prop osition 2.3 (1 ) . If h = 0, it is true for K h = { x } for an y x ∈ X . Now w e assume 0 < h < h top ( T , X ). By the v ariational principle there exists µ ∈ M e ( X , T ) with h < h µ ( T ) ≤ h top ( T , X ) < + ∞ . It is well kno wn [14, Theorem 15.11 ] that there exists a T -in v aria n t sub- σ - algebra C ⊆ B µ suc h t ha t h µ ( T , X |C ) = h , where B µ is the completion of B X under µ . Then the conclusion f o llo ws fro m Lemma 4 .3 (2).  Let ( X , T ) b e a TDS, µ ∈ M ( X , T ) and C ⊆ B µ a T - in v arian t sub- σ -algebra with µ = R X µ x dµ ( x ) the disin tegratio n of µ o v er C . F o r µ - a.e. x ∈ X , w e define h B U ( T , µ, x ) = inf { h B U ( T , Z ) : Z ∈ B X with µ x ( Z ) = 1 } for an y giv en U ∈ C X and h B ( T , µ, x ) = sup U ∈C o X h B U ( T , µ, x ) . The essential supr emum of a real v alued function f de fined on a subset of X with µ -full measure is defined by µ − sup f ( x ) = inf µ ( X ′ )=1 sup x ∈ X ′ f ( x ) . W en Hu ang, Xiangdong Y e and Guoh ua Zhang 15 W e are not sure of the µ - measurabilit y of functions b oth h B U ( T , µ, x ) and h B ( T , µ, x ) w.r.t. x ∈ X . Whereas, using Prop osition 4.2 we hav e the following result. Corollary 4.5. L et ( X , T ) b e a TDS , µ ∈ M ( X, T ) a n d C ⊆ B µ a T -invariant sub- σ -algebr a. I f µ = R X µ x dµ ( x ) is the disinte gr ation of µ over C , then (1) L e t U ∈ C X and α ∈ P X with f C T ,α ( x ) the function ob taine d in The or em 4.1 for T , α ∈ P X and C . Assume that e ach element o f U has a non-empty interse ction with at most M elemen ts of α ( M ∈ N ). Then for µ - a .e. x ∈ X , h B U ( T , µ, x ) ≥ f C T ,α ( x ) − lo g M and if µ is er go d i c then h B U ( T , µ, x ) ≥ h µ ( T , α |C ) − log M . (2) µ − sup h B ( T , µ, x ) ≥ h µ ( T , X |C ) ; mor e over, if µ is er go dic then h B ( T , µ, x ) = h µ ( T , X |C ) for µ -a.e. x ∈ X . Pr o of. (1) is just a direct corollary of Prop osition 4.2 (1). (2) F or k ∈ N we tak e U k ∈ C o X with || U k || ≤ 1 k . Then, for eac h n ∈ N , w e tak e α n,k ∈ P X suc h that α n,k  ( U k ) n − 1 0 and at most n # U k elemen ts of α n,k can hav e a p oin t in all their closures, a nd tak e U n,k ∈ C o X suc h t ha t eac h elemen t of U n,k has a non-empt y interse ction with at most n # U k elemen ts of α n,k . Let f C T n ,α n,k ( x ) b e t he function obtained in Theorem 4.1 for T n , α n,k and C . Then f C T n ,α n,k ( x ) is T n -in v a r ia n t and R X f C T n ,α n,k ( x ) dµ ( x ) = h µ ( T n , α n,k |C ). Then using (1 ), for µ -a.e. x ∈ X , nh B U n,k ( T , µ, x ) = inf { nh B U n,k ( T , Z ) : Z ∈ B X with µ x ( Z ) = 1 } ≥ inf { h B U n,k ( T n , Z ) : Z ∈ B X with µ x ( Z ) = 1 } (by Prop osition 2.3 (4 )) ≥ f C T n ,α n,k ( x ) − log ( n # U k ) (using (1)) . Hence, µ − sup h B ( T , µ, x ) ≥ µ − sup h B U n,k ( T , µ, x ) ≥ 1 n Z X  f C T n ,α n,k ( x ) − log ( n # U k )  dµ ( x ) = 1 n ( h µ ( T n , α n,k |C ) − log( n # U k )) ≥ 1 n ( h µ ( T n , ( U k ) n − 1 0 |C ) − log ( n # U k )) = h µ ( T , U k |C ) − 1 n log( n # U k ) . Fixing k ∈ N letting n → + ∞ in the ab ov e inequalit y w e obtain µ − sup h B ( T , µ, x ) ≥ h µ ( T , U k |C ). Then letting k → + ∞ w e obtain µ − sup h B ( T , µ, x ) ≥ h µ ( T , X |C ). No w w e a ssum e tha t µ is ergo dic. F irst, by Prop osition 4.2 (2), w e kno w h B ( T , µ, x ) ≥ h µ ( T , X |C ) for µ -a.e. x ∈ X . Sec ondly using Lemm a 4.3 (2), there exists a measurable subset X ′ of X with µ ( X ′ ) = 1 suc h that if x ∈ X ′ then for eac h ℓ ∈ N there exists a compact s ubset Z x ( ℓ ) of X suc h that µ x ( Z x ( ǫ )) ≥ 1 − 1 2 ℓ and h B ( T , Z x ( ℓ )) = h µ ( T , X |C ). Next for eac h x ∈ X ′ , let Z x = S ℓ ∈ N Z x ( ℓ ). Then Z x ∈ B X with µ ( Z x ) = 1 and h B ( T , Z x ) = sup ℓ ∈ N h B ( T , Z x ( ℓ )) = h µ ( T , X |C ). This 16 lo weri ng top olog ical entrop y ov er subsets implies h B ( T , µ, x ) ≤ h µ ( T , X |C ). Collecting terms, h B ( T , µ, x ) = h µ ( T , X |C ) for µ -a.e. x ∈ X .  F ollowing from the pro of of Corollary 4.5, w e are easy to show the follo wing result. Corollary 4.6. L et ( X , T ) b e a TDS, µ ∈ M e ( X , T ) and C ⊆ B µ a T -invariant sub- σ -algebr a. I f µ = R X µ x dµ ( x ) is the disinte gr ation of µ over C , then (1) If α ∈ P X then for µ -a.e. x ∈ X ther e exi s ts Z x ∈ B X such that µ x ( Z x ) = 1 and h B α ( T , Z x ) = h µ ( T , α |C ) . Mor e over, h B α ( T , µ, x ) = h µ ( T , α |C ) for µ -a.e. x ∈ X . (2) F or µ -a.e. x ∈ X ther e exi s ts Z x ∈ B X such that µ x ( Z x ) = 1 and h B ( T , Z x ) = h µ ( T , X |C ) . Remark 4.7. We c an ’t exp e ct simil a r r esults ho l d fo r top olo gic al entr opy of subsets using op en c overs. F or e x ample, let ( X , T ) b e a minimal TDS, µ ∈ M e ( X , T ) and C = {∅ , X } such that 0 < h µ ( T , X ) < h top ( T , X ) . L et µ = R X µ x dµ ( x ) b e the di s i nte gr a tion o f µ o ver C , then µ x = µ for µ -a. e . x ∈ X . Thus for µ -a.e. x ∈ X , if Z ∈ B X with µ x ( Z ) = 1 then Z = X , which implies h ( T , Z ) = h ( T , Z ) = h top ( T , X ) > h µ ( T , X ) = h µ ( T , X |C ) . 5. Exp ansive cases In this section b y direct construction we shall prov e t ha t each expansiv e TDS is HUL . Recall that w e say a TDS ( X , T ) is ex p ansive if t here exists δ > 0 suc h that x 6 = y implies sup n ∈ Z d ( T n x, T n y ) > δ . In t his case, δ is called an exp ansive c onstant . In particular, each sym b olic TDS is expansiv e. T o do this let’s first recall [30, R emark 5.13]. Let ( X , T ) b e a TDS with metric d and E a compact subset. F o r each ǫ > 0 and x ∈ E w e define h d ( x, ǫ, E ) = inf { r ( d, T , ǫ, K ) : K is a compact neigh b orho o d of x in E } . Let h ( x, E ) = lim ǫ → 0+ h d ( x, ǫ, E ). Its v alue dep ends only on the to po lo gy on X . The f ollo wing is [30, Remark 5 .1 3]. Theorem 5.1. L et ( X, T ) b e a T D S with metric d and E a c omp act subset. Then (1) h d ( x, ǫ, E ) is u.s.c. on E and sup x ∈ E h ( x, E ) = h ( T , E ) . (2) F or e ach x ∈ E ther e is a c ountable c omp act subset E x ⊆ E with a unique limit p oi n t x such that h ( T , E x ) = h ( x, E ) . (3) Ther e is a c ountable c omp a ct subset E ′ ⊆ E with h ( T , E ′ ) = h ( T , E ) . Mor e- over, E ′ c an b e chosen such that the set of its limit p oints has at most one limit p oint, and E ′ has a unique limit p oint iff ther e is x ∈ E with h ( x, E ) = h ( T , E ) . The first result is the follo wing lemma. Lemma 5.2. L et ( X , T ) b e a TDS with metric d and K ⊆ X a c omp act subset with h ( T , K ) > 0 . Then for any 0 < h < h ( T , K ) ther e is a δ 0 > 0 such that if 0 < δ ≤ δ 0 then ther e is a c o untable c omp ac t subset K h,δ ⊆ K with a unique limit p oint such that s ( d, T , δ , K h,δ ) = h . W en Hu ang, Xiangdong Y e and Guoh ua Zhang 17 Pr o of. Let 0 < h < h ( T , K ). By Theorem 5.1 there exists a coun table compact subset K 0 ⊆ K with a unique limit p oint x 0 suc h t ha t h ( T , K 0 ) > h , th us for some δ 0 > 0 if 0 < δ ≤ δ 0 then s ( d, T , δ, K 0 ) > h . No w let 0 < δ ≤ δ 0 b e fixed. Define l 1 to b e the minimal in teger n ∈ N suc h that ∃ B 1 ⊆ K 0 is ( n, δ )- separated w.r.t. T s.t. | B 1 | = [ e nh ] + 2 , here | B 1 | means the cardinalit y o f B 1 . It is clear tha t l 1 is finite, as s ( d, T , δ , K 0 ) > h . Let A 1 = D 1 ⊆ K 0 b e ( l 1 , δ )- separated w.r.t. T with | A 1 | = [ e l 1 h ] + 1 and x 0 / ∈ A 1 . Define l 2 to b e the minimal in teger n > l 1 suc h that ∃ B 2 ⊆ ( B d l 1 ( x 0 , δ ) ∩ K 0 ) \ A 1 is ( n, δ )-separated w.r.t. T s.t. | B 2 | = [ e nh ] − [ e l 1 h ] + 2 , where B d l 1 ( x 0 , δ ) denotes the op en ba ll with cen ter x 0 and radius δ (in the sense of d l 1 -metric). Since x 0 is the uniq ue limit p oin t of the countable compact subset K 0 ⊆ K , K 0 \ (( B d l 1 ( x 0 , δ ) ∩ K 0 ) \ A 1 ) is a finite subset, so s ( d, T , δ, ( B d l 1 ( x 0 , δ ) ∩ K 0 ) \ A 1 ) = s ( d, T , δ , K 0 ) > h, whic h implies t ha t l 2 > l 1 is finite. Let D 2 ⊆ ( B d l 1 ( x 0 , δ ) ∩ K 0 ) \ A 1 b e ( l 2 , δ )- separated w.r.t. T with | D 2 | = [ e l 2 h ] − [ e l 1 h ] + 1 and x 0 / ∈ D 2 . Set A 2 = A 1 ∪ D 2 6∋ x 0 . Then | A 2 | = [ e l 2 h ] + 2 and A 2 ⊆ K 0 is ( l 2 , δ )- separated w.r.t. T . By induction there are l 1 < l 2 < · · · and A 1 ⊆ A 2 ⊆ · · · ⊆ K 0 suc h t ha t f o r each i ∈ N (1) x 0 6∈ A i and A i +1 \ A i ⊆ B d l i ( x 0 , δ ) ∩ K 0 . (2) A i is ( l i , δ )- separated w.r.t. T and | A i | = [ e l i h ] + i . Set A ∞ = { x 0 } ∪ S i ≥ 1 A i ⊆ K 0 . Then A ∞ ( ⊆ K 0 ) ⊆ K is a coun table compac t subset with x 0 as its unique limit p oin t in X . If l n ≤ l < l n +1 then let A ⊆ A ∞ b e ( l , δ )-separated w.r.t. T . As A \ A n ⊆ ( B d l n ( x 0 , δ ) ∩ K 0 ) \ A n is ( l, δ )-separated w.r.t. T , b ecause o f the definition of l n +1 w e hav e | A \ A n | ≤ [ e lh ] − [ e l n h ] + 1, whic h implies | A | ≤ | A \ A n | + | A n | ≤ ([ e lh ] − [ e l n h ] + 1) + ([ e l n h ] + n ) = [ e lh ] + n + 1 . Then s l ( d, T , δ , A ∞ ) ≤ [ e lh ] + n + 1 for a ll l n ≤ l < l n +1 . Note that s l n ( d, T , δ , A ∞ ) ≥ [ e l n h ] + n , w e conclude s ( d, T , δ, A ∞ ) = h . T ak e K h,δ = A ∞ . This completes the pro of.  Before pro ving that eac h expansiv e TDS is H UL we need the fo llowing r esult. Lemma 5.3. L et ( X , T ) b e an exp ansive TDS with metric d and an exp a nsive c on- stant δ > 0 . Then for any c o mp act subset K ⊆ X , h ( T , K ) = s ( d, T , δ 2 , K ) . Pr o of. Let K ⊆ X b e a compact subset and ǫ > 0. W e claim that there exists n ( ǫ ) ∈ N suc h that for x, y ∈ X , d ∗ n ( ǫ ) ( x, y ) ≤ δ 2 implies d ( x, y ) ≤ ǫ , where d ∗ n ( ǫ ) ( x, y ) = max n ( ǫ ) i = − n ( ǫ ) d ( T i x, T i y ). In fact, if it is not the case, then for eac h n ∈ N t here exist x n , y n ∈ X suc h t hat d ∗ n ( x n , y n ) ≤ δ 2 and d ( x n , y n ) > ǫ . W.l.g. we assume lim n → + ∞ ( x n , y n ) = ( x, y ). Then d ( x, y ) ≥ ǫ and d ∗ m ( x, y ) ≤ δ 2 for each m ∈ N , whic h con tradicts that δ is a n expansiv e constant of ( X , T ). 18 lo weri ng top olog ical entrop y ov er subsets No w for each m ∈ N , let E b e an ( m, ǫ )-separated subset of K w.r.t. T , then T − n ( ǫ ) E is an ( m + 2 n ( ǫ ) , δ 2 )-separated subset of T − n ( ǫ ) K w.r.t. T . Hence s ( d, T , ǫ, K ) = lim sup m → + ∞ 1 m log s m ( d, T , ǫ, K ) ≤ lim sup m → + ∞ 1 m log s m +2 n ( ǫ )  d, T , δ 2 , T − n ( ǫ ) K  = s  d, T , δ 2 , T − n ( ǫ ) K  = s  d, T , δ 2 , K  . Sine ǫ > 0 is arbitrary , letting ǫ → 0+ we conclude h ( T , K ) = s ( d, T , δ 2 , K ).  No w we are ready to prov e the main result in this section. Theorem 5.4. Each exp ansive TDS is HUL . Pr o of. Let ( X , T ) b e an expansiv e TDS with metric d and an expansiv e constant 2 δ > 0. Let E ⊆ X b e a no n-empt y compact subset and 0 ≤ h ≤ h ( T , E ). If h = 0 then h ( T , { x } ) = 0 fo r any x ∈ E . Now assume h = h ( T , E ) > 0, then by Theorem 5 .1 (1), h d ( x, δ, E ) is u.s.c. on E and h ( T , E ) = sup x ∈ X h ( x, E ). Note tha t h ( x, E ) = lim ǫ → 0+ inf { r ( d, T , ǫ, K ) : K is a compact neigh b orho o d of x in E } = lim ǫ → 0+ inf { s ( d, T , ǫ, K ) : K is a compact neighbor ho o d of x in E } = h d ( x, δ, E ) (using Lemma 5.3) , then h ( T , E ) = max x ∈ E h d ( x, δ, E ) . Sa y x 0 ∈ E with h d ( x 0 , δ, E )(= h ( x 0 , E )) = h ( T , E ) > 0. By Theorem 5.1 (2 ) there exists a coun table compact subset K 0 ⊆ E with x 0 as its unique limit p oin t suc h that h ( T , K 0 ) = h ( x 0 , E ) = h ( T , E ). This completes the pro of in the case of h = h ( T , E ) > 0. No w assume 0 < h < h ( T , E ). By Lemma 5.2 there exist 0 < ǫ ≤ δ small enough and a coun table compact subset K h ⊆ E with a unique limit p oint suc h that s ( d, T , ǫ, K h ) = h . Since 2 ǫ is a lso a n expansiv e constan t, h ( T , K h ) = h by Lemma 5.3. Th us fo r each non-empt y compact subset E and eac h 0 ≤ h ≤ h ( T , E ) there is a non-empt y compact subset K h ⊆ K with a unique limit p oin t suc h that h ( T , K h ) = h , that is, TDS ( X , T ) is HUL .  Note that w e can also in tro duce the HUL prop ert y for a GTDS, the r esults of this section remain true f or a GTD S. In particular, following similar discussions, the results in Lemma 5.3 and Theorem 5.4 also hold for eac h p ositiv ely expansiv e dynamical system. Let X b e a compact metric space endow ed with a contin uo us sur- jection T : X → X and a compatible metric d . Recall that we say ( X , T ) p ositively exp a nsive if there exists δ > 0 suc h that x 6 = y implies sup n ∈ Z + d ( T n x, T n y ) > δ (see for example [27]). W e a lso call δ an exp ansive c onstant . W en Hu ang, Xiangdong Y e and Guoh ua Zhang 19 6. A HUL TDS is as ymptoticall y h-exp ansive In this section w e shall answ er Question 1.4 partially . Not e that the inv ertibility can b e remov ed for TDSs considered in this section without c hanging our results. W e discuss t w o classes of w eak expansiv enes s: the h -expansiv eness a nd a sym p- totical h -expansiv eness, in tro duced by Bow en [3] and Misiurewicz [24], r esp ectiv ely . Let ( X , T ) b e a GTDS with metric d . F or eac h ǫ > 0 we define h ∗ T ( ǫ ) = sup x ∈ X h ( T , Φ ǫ ( x )) , where Φ ǫ ( x ) = { y ∈ X : d ( T n x, T n y ) ≤ ǫ if n ≥ 0 } . ( X , T ) is called h -exp an sive if there exists an ǫ > 0 suc h that h ∗ T ( ǫ ) = 0, and is called asymptotic al ly h -exp ansive if lim ǫ → 0+ h ∗ T ( ǫ ) = 0. It is sho wn by Bo w en [3] that p osi- tiv ely expansiv e systems , expansiv e homeomorphisms, endomorphisms of a compact Lie g r oup and Axiom A diffeomorphisms are all h - expansiv e, b y Misiurewicz [25] that ev ery con tinu ous endomorphism of a compact metric group is asymptotically h -expansiv e if it has finite entrop y , and b y Buz zi [7] that any C ∞ diffeomorphism on a compact manifold is asymptotically h -expansiv e. In this section we prov e that eac h H UL TDS is asymptotically h -expansiv e. The follo wing t w o r esults seem to o tec hnical but interesting themselv es, whic h are needed in prov ing the main result of this section. Theorem 6.1. L et ( X , T ) b e a TDS. Then for a n y c omp act subset K ⊆ X with h ( T , K ) > 0 , ther e is a c ountable in finite c omp a ct subset K ∞ ⊆ K such that h ( T , K ∞ ) = 0 . Pr o of. First, there is a countable compact subset K 0 = { x, x 1 , x 2 , · · · } ⊆ K such that h = h ( T , K 0 ) > 0 and lim n → + ∞ x n = x (using Theorem 5.1) . Let d b e a metric on ( X , T ). F or sufficien tly small ǫ 1 > 0 let K 1 ⊆ K 0 b e the subset constructed in Lemma 5.2 s uc h that s ( d, T , ǫ 1 , K 1 ) = h 2 . No w if K n , n ∈ N , is constructed, for a more smaller 0 < ǫ n +1 < ǫ n , b y Lemma 5.2 w e let K n +1 b e a prop er compact subset o f K n with s ( d , T , ǫ n +1 , K n +1 ) = h n +2 . In fact, w e can r equire that lim n → + ∞ ǫ n = 0. Now let K ∞ = { x, y 1 , y 2 , · · · } b e a subset of K 0 , where y n ∈ K n \ K n +1 for eac h n ∈ N . It is clear that K ∞ ⊆ K is a coun table infinite compact subset, and s ( d, T , ǫ n , K ∞ ) ≤ s ( d, T , ǫ n , K n ) ≤ h n for eac h n ∈ N , as K ∞ \ { y 1 , · · · , y n − 1 } ⊆ K n . Hence w e hav e h ( T , K ∞ ) = lim n → + ∞ s ( d, T , ǫ n , K ∞ ) = 0 . This completes the pro of.  Lemma 6.2. L et ( X, T ) b e a TDS. Assume that { B n } n ∈ N ⊆ 2 X satisfies lim n → + ∞ B n = { x 0 } (in the s e nse o f Hausdorff metric) for some x 0 ∈ X and (6.1) inf J ∈ Z + lim n → + ∞ sup j ≥ J diam ( T j B n ) = 0 . 20 lo weri ng top olog ical entrop y ov er subsets L et x n ∈ B n for e ac h n ∈ N . Then (6.2) h T , + ∞ [ n =1 B n ∪ { x 0 } ! = max  sup n ∈ N h ( T , B n ) , h ( T , { x i } ∞ 0 )  . If in addition x 0 6∈ B n for e ach n ∈ N , then any c ountable c omp act subset of S + ∞ n =1 B n ∪ { x 0 } with a unique lim it p oint x 0 has entr opy at most h ( T , { x i } ∞ 0 ) . Pr o of. Let d b e a metric on ( X , T ) a nd ǫ > 0. Th us by (6.1) there exist i ǫ , j ǫ ∈ N suc h that if i ≥ i ǫ then ǫ i < ǫ 4 , where ǫ i = sup j ≥ j ǫ diam( T j B i ). Set X 1 = T j ǫ ( S i ∈ N B i ∪ { x 0 } ). F or n ∈ N , let E n b e an ( n, ǫ )-separated subset of X 1 w.r.t. T with s n ( d, T , ǫ, X 1 ) = | E n | . It is clear that if i ≥ i ǫ then | E n ∩ T j ǫ B i | ≤ 1, and if y ∈ E n ∩ T j ǫ B i then d n ( y , T j ǫ x i ) ≤ ǫ i < ǫ 4 . Say E n ∩ T j ǫ ( S i ≥ i ǫ B i ) = { y 1 , · · · , y l } . F or eac h 1 ≤ r ≤ l , there exists i r ≥ i ǫ suc h that y r ∈ E n ∩ T j ǫ B i r . Let F n = { T j ǫ x i 1 , · · · , T j ǫ x i r } . F or eac h 1 ≤ r 1 < r 2 ≤ l , d n ( T j ǫ x i r 1 , T j ǫ x i r 2 ) ≥ d n ( y r 1 , y r 2 ) − ( d n ( y r 1 , T j ǫ x i r 1 ) + d n ( y r 2 , T j ǫ x i r 2 )) > ǫ 2 . Hence, F n is an ( n, ǫ 2 )-separated subset of T j ǫ ( { x i } ∞ 0 ) w.r.t. T , whic h implies s n ( d, T , ǫ 2 , T j ǫ ( { x i } ∞ 0 )) ≥ l . Not e that G n . = E n ∩ T j ǫ ( S i ǫ − 1 i =1 B i ) is an ( n, ǫ )-separated subset of T j ǫ ( S i ǫ − 1 i =1 B i ) w.r.t. T , w e hav e s n ( d, T , ǫ, X 1 ) = | E n | ≤ | G n ∪ { y 1 , · · · , y l } ∪ { T j ǫ x 0 }| ≤ | G n | + l + 1 ≤ s n d, T , ǫ, T j ǫ i ǫ − 1 [ i =1 B i !! + s n  d, T , ǫ 2 , T j ǫ ( { x i } ∞ 0 )  + 1 , whic h implies that s d, T , ǫ, [ i ∈ N B i ∪ { x 0 } ! = s ( d, T , ǫ, X 1 ) ≤ max ( s d, T , ǫ, T j ǫ i ǫ − 1 [ j =1 B j !! , s  d, T , ǫ 2 , T j ǫ ( { x i } ∞ 0 )  ) = max ( s d, T , ǫ, i ǫ − 1 [ j =1 B j ! , s  d, T , ǫ 2 , { x i } ∞ 0  ) ≤ max  max 1 ≤ j ≤ i ǫ − 1 h ( T , B i ) , h ( T , { x i } ∞ 0 )  . Th us w e o btain the direction ” ≤ ” o f (6.2). The other direction is o b vious. If in additional x 0 6∈ B n for eac h n ∈ N , let K ⊆ S + ∞ n =1 B n ∪ { x 0 } b e an y coun table compact subset with a unique limit p oint x 0 . Set B ′ n = K ∩ B n . Then B ′ n is finite for each n ∈ N , as x 0 is the unique limit p oin t of K and x 0 6∈ B ′ n . So we hav e W en Hu ang, Xiangdong Y e and Guoh ua Zhang 21 { B ′ n ∪ { x n }} n ∈ N ⊆ 2 X and h ( T , K ) ≤ h ( T , K ∪ { x i } ∞ i =0 ) = h T , + ∞ [ n =1 ( B ′ n ∪ { x n } ) ∪ { x 0 } ! = max  sup n ∈ N h ( T , B ′ n ∪ { x n } ) , h ( T , { x i } ∞ 0 )  (using (6.2)) = h ( T , { x i } ∞ 0 ) (as B ′ n ∪ { x n } is a finite subset for eac h n ∈ N ) . This finishes the pro of.  Remark 6.3. Without the assumption of (6.1) , in gene r al L emma 6.2 d o esn ’t hold. F or example, let { x n } n ∈ N b e a sequence of X with limit x ∈ X and h ( T , { x, x 1 , x 2 , · · · } ) = a > 0. By Theorem 6 .1 there is a sub-sequence { n i } i ∈ N suc h t ha t h ( T , { x, x n 1 , x n 2 , · · · } ) = 0. Let B j = { x n j − 1 , x n j − 1 +1 , · · · , x n j − 1 } for each j ∈ N , whe re n 0 = 1. Then lim j → + ∞ diam( B j ) = 0 and h ( T , S j ∈ N B j ∪ { x } ) = a > 0, but sup j ∈ N h ( T , B j ) + h ( T , { x, x n 1 , x n 2 , · · · } ) = 0 . In fact, for a go o d c hoice of the sub-sequence { n i } i ∈ N in the ab o v e construction, w e can require that h ( T , S i ∈ N B k i ∪ { x } ) = a > 0 for an y sub-sequence { k i } i ∈ N ⊆ N . This is done as follows . F or eac h j ∈ N w e can select a sub-seque nce { m j k } k ∈ N ⊆ N suc h that lim k → + ∞ log s m j k ( d, T , 1 j , { x l : l ≥ n i } ) m j k = s  d, T , 1 j , { x l : l ≥ n i }   = s  d, T , 1 j , { x, x 1 , x 2 , . . . }  for each i ∈ N . W e may assume ( r eplace the sequences { n i } i ∈ N and { m j k } k ∈ N b y sub-sequenc es if necessary) s m j i  d, T , 1 j , { x l : n i ≤ l < n i +1 }  ≥ e m j i ( s ( d,T , 1 j , { x,x 1 ,x 2 , ··· } ) − 1 i ) if 1 ≤ j ≤ i. Let B j = { x n j − 1 , x n j − 1 +1 , · · · , x n j − 1 } . Then sup j ∈ N h ( T , B j ) + h ( T , { x, x n 1 , x n 2 , · · · } ) = 0 . No w for any sub-sequence { k i } i ∈ N ⊆ N w e hav e: if l ∈ N and 1 ≤ j ≤ k l then s m j k l d, T , 1 j , [ i ∈ N B k i ∪ { x } ! ≥ s m j k l  d, T , 1 j , B k l  ≥ e m j k l ( s ( d,T , 1 j , { x,x 1 ,x 2 , ··· } ) − 1 k l ) , 22 lo weri ng top olog ical entrop y ov er subsets whic h implies that fo r each fixed j ∈ N s  d, T , 1 j , { x, x 1 , x 2 , · · · }  ≥ s d, T , 1 j , [ i ∈ N B k i ∪ { x } ! ≥ lim sup l → + ∞ 1 m j k l log s m j k l d, T , 1 j , [ i ∈ N B k i ∪ { x } ! ≥ s  d, T , 1 j , { x, x 1 , x 2 , · · · }  . Then letting j → + ∞ w e ha v e h ( T , S i ∈ N B k i ∪ { x } ) = h ( T , { x, x 1 , x 2 , · · · } ) = a . No w we are ready to prov e the main result in this section. Theorem 6.4. Each HUL TDS is asymptotic al ly h-exp ansive. Pr o of. Let ( X , T ) b e a HUL TD S with metric d . Assume the contrary that ( X , T ) is not asymptotically h -expansiv e, i.e. a = h ∗ ( T ) = lim ǫ → 0+ h ∗ T ( ǫ ) > 0 . Then there exist a sequence { x i } i ∈ N ⊆ X with limit x and a sequence { ǫ i } i ∈ N of p ositiv e n um b ers with limit 0 suc h that lim i → + ∞ h ( T , Φ ǫ i ( x i )) = a . There are t w o cases. Case 1. There exis t infinitely many i ∈ N suc h that for whic h x 6∈ Φ ǫ i ( x i ). Th us w.l.g. w e assume x 6∈ Φ ǫ i ( x i ) f or eac h i ∈ N . Since ( X , T ) is HUL , for eac h i ∈ N w e can ta ke a coun table infinite compact subset X i ⊆ Φ ǫ i ( x i ) with a unique limit p oin t y i suc h that a i . = h ( T , X i ) < a and lim i → + ∞ a i = a . Clearly , lim i → + ∞ y i = x . Moreov er, by Theorem 6 .1 w e ma y assume that h ( T , { x, y 1 , y 2 , · · · } ) = 0 (if necessary w e ta k e a sub-sequence). By Lemma 6.2, h ( T , S i ∈ N X i ∪ { x } ) = max { sup i ∈ N h ( T , X i ) , h ( T , { x, y 1 , y 2 , · · · } ) } = a , so there exists a countable infinite compact subset A ⊆ S i ∈ N X i ∪ { x } with a unique limit p oint z suc h that h ( T , A ) = a > 0. By a ssumptions, if z = x then each A i . = A ∩ X i is a finite subset, whic h implies h ( T , A ) ≤ h ( T , A ∪ { y 1 , y 2 , · · · } ) = max  sup i ∈ N h ( T , A i ∪ { y i } ) , h ( T , { x, y 1 , y 2 , · · · } )  (using Lemma 6.2) = 0 , a contradiction with h ( T , A ) = a > 0. Th us z 6 = x , and so z ∈ X v for some v ∈ N . Put r = d ( z ,x ) 2 , then r > 0 follows from x / ∈ Φ ǫ v ( x v ). Since { x } is the limit of { X i } i ∈ N (in the sense of Hausdorff metric H d ), there exists L ∈ N suc h that if i > L then d ( z , X i ) ≥ d ( z , x ) − H d ( X i , { x } ) > r . Th us ( S i>L X i ∪ { x } ) ∩ A is a finite set, as z is the unique limit p oint of A and d ( z , ( S i>L X i ∪ { x } ) ∩ A ) ≥ r . Therefore h ( T , A ) = h T , A ∩ [ 1 ≤ i ≤ L X i ! ≤ max 1 ≤ i ≤ L h ( T , X i ) = max 1 ≤ i ≤ L a i < a, a con tradiction. Case 2. x ∈ Φ ǫ i ( x i ) f or eac h la r ge enough i ∈ N . W.l.g. w e a ssume that fo r eac h i ∈ N , x ∈ Φ ǫ i ( x i ) a nd so Φ ǫ i ( x i ) ⊆ Φ 2 ǫ i ( x ). So lim ǫ → 0+ h ( T , Φ ǫ ( x )) = a . As h ( T , Φ ǫ ( T k x )) ≥ h ( T , Φ ǫ ( x )) for eac h k ∈ N and ǫ > 0, then lim ǫ → 0+ h ( T , Φ ǫ ( T k x )) = a for eac h k ∈ N . W en Hu ang, Xiangdong Y e and Guoh ua Zhang 23 If { x, T x, T 2 x, · · · } is infinite, w e fix a p oin t y ∈ ω ( x, T ) . = T n ∈ N { T j x : j ≥ n } . Then f o r eac h i ∈ N there exists k i ∈ N such that 0 < d ( y , T k i x ) < 1 i . F or eac h i ∈ N , as lim ǫ → 0+ h ( T , Φ ǫ ( T k i x )) = a , we may tak e 0 < η i < d ( y , T k i x ) suc h that h ( T , Φ η i ( T k i x )) > min { a (1 − 1 i ) , i } . Let y i = T k i x , then lim i → + ∞ y i = y and lim i → + ∞ h ( T , φ η i ( y i )) = a and y / ∈ Φ η i ( y i ) for eac h i ∈ N . By a similar pro of to Case 1, it is imp ossible. Hence, x m ust ha v e a finite orbit, w e may assu me tha t x is a p erio dic p oint (if necessary w e replace x b y T k x for some k ∈ N ). Let l ∈ N b e the p eriod o f x . Since T (Φ ǫ ( T k x )) ⊆ Φ ǫ ( T k +1 x ) for eac h k ∈ N and ǫ > 0, S l − 1 i =0 Φ ǫ ( T i x ) is compact and T - in v ariant (i.e. T ( S l − 1 i =0 Φ ǫ ( T i x )) ⊆ S l − 1 i =0 Φ ǫ ( T i x )) for any ǫ > 0 . F or eac h n ∈ N , let Y n = S l − 1 i =0 Φ 1 n ( T i x ). The n ( Y n , T ) is a sub-system of ( X , T ) and h top ( T , Y n ) ≥ h ( T , Φ 1 n ( x )) ≥ a . By the v ariational principle, t here exists µ n ∈ M e ( Y n , T ) suc h that h µ n ( T , Y n ) > min { a (1 − 1 n ) , n } . Ob viously , µ n ( { x } ) = 0 and µ n (Φ 1 n ( x )) ≥ 1 l . Th us, w e can tak e a compact subset K n ⊆ Φ 1 n ( x ) suc h that µ n ( K n ) > 0 and x / ∈ K n . By a classic result of Kat o k [16, Theorem 1.1] (see also [30, Theorem 3 .7]), w e kno w that h ( T , K n ) ≥ h µ n ( T , Y n ) > min  a  1 − 1 n  , n  , as µ n ( K n ) > 0 . As ( X , T ) is HUL , there is a coun ta ble compact subset X n ⊆ K n with a unique limit point y n suc h that h ( T , X n ) = a n = min { a (1 − 1 n ) , n } < a . Clearly , x / ∈ X n for eac h n ∈ N , as x / ∈ K n . Again b y a similar pro o f to Case 1 , w e kno w that this is imp ossible. Th us, ( X , T ) m ust b e asymptotically h - expans iv e.  7. Pr oper ties preser ved by a princip al extension Com bined with the results obtained in sections 5 and 6, in this section w e shall an- sw er question 1.4 b y proving that a TDS is HUL iff it is asymptotically h -expansiv e. Moreo v er, we presen t a hereditarily lo w erable TDS with finite entrop y whic h is not HUL . As a b ypro duct, w e show that principal extension preserv es the prop erties of lo w ering, hereditary low ering and HUL . Let ( X , T ) and ( Y , S ) be GTDSs. W e say that π : ( X , T ) → ( Y , S ) is a factor map if π is a con tin uous surjectiv e map and π ◦ T = S ◦ π . Let π : ( X , T ) → ( Y , S ) b e a factor map betw een GTDSs. The r e lative top o l o gic al entr opy of ( X, T ) w.r.t. π is defined as fo llo ws: h top ( T , X | π ) = sup y ∈ Y h ( T , π − 1 ( y )) . Let π : ( X , T ) → ( Y , S ) b e a factor map b et w een GTDSs. It’s easy to che c k that on Y the function y 7→ h ( T , π − 1 ( y )) is S -in v ariant and Borel measurable. Th us for eac h ν ∈ M ( Y , S ) we may define h ( T , X | ν ) = Z Y h ( T , π − 1 ( y )) dν ( y ) . In particular, if ν is ergo dic then h ( T , π − 1 ( y )) = h ( T , X | ν ) for ν -a.e y ∈ Y . Thus 24 lo weri ng top olog ical entrop y ov er subsets Prop osition 7.1. L et π : ( X , T ) → ( Y , S ) b e a factor map b etwe en GTDSs. Then for e ach ν ∈ M e ( Y , S ) ther e exists a c ountable c omp act se t K ∈ 2 X with h ( T , K ) = h ( T , X | ν ) . No w let π : ( X , T ) → ( Y , S ) b e a factor map b et w een GTDSs and µ ∈ M ( X, T ), note that t he sub- σ - a lgebra π − 1 ( B Y ) ⊆ B X satisfies T − 1 π − 1 ( B Y ) ⊆ π − 1 ( B Y ) in the sense of µ , w e define r elative me asur e-the or etic al µ -entr opy of ( X , T ) w. r.t. π as h µ ( T , X | π ) = h µ ( T , X | π − 1 ( B Y )) . The following results ar e prov ed in [10] a nd [19]. Lemma 7.2. L et π : ( X , T ) → ( Y , S ) b e a factor map b etwe en GTDSs. Then (1) One has h top ( T , X | π ) = sup ν ∈M ( Y ,S ) h ( T , X | ν ) = sup ν ∈M e ( Y ,S ) h ( T , X | ν ) . (2) F or e ach ν ∈ M ( Y , S ) , h ( T , X | ν ) = sup { h µ ( T , X | π ) : µ ∈ M ( X, T ) , π µ = ν } . (3) F or e ach µ ∈ M ( X , T ) , h µ ( T , X ) = h µ ( T , X | π ) + h π µ ( S, Y ) . W e hav e pro v ed that each expansiv e TDS is HUL . In fact, t he same conclu sion holds for a more g eneral case . T o prov e this, first w e shall prov e the follo wing Bo w en’s type t heorem whic h is in teresting itself. W e remark that the idea of t he pro of is inspired b y the pro of o f [2 , Theorem 17]. Theorem 7.3. L et π : ( X , T ) → ( Y , S ) b e a factor m ap b etwe en GTDS s and E ∈ 2 X . Then h ( S, π ( E )) ≤ h ( T , E ) ≤ h ( S, π ( E )) + h top ( T , X | π ) . In p articular, if ( Y , S ) has finite entr op y then for e a c h E ∈ 2 X , h ( T , E ) − h ( S, π ( E )) ≤ h top ( T , X | π ) . Pr o of. By the contin uity of π , it’s easy to obtain h ( S, π ( E )) ≤ h ( T , E ). Th us it remains to pro v e h ( T , E ) ≤ h ( S, π ( E )) + h top ( T , X | π ) . If h top ( T , X | π ) = + ∞ , this is obvious . No w w e supp ose that a = h top ( T , X | π ) < + ∞ . Let d X and d Y b e the metrics on ( X , T ) and ( Y , S ), resp ectiv ely . Let ǫ > 0 and α > 0. By Lemma 7.2 ( 1), for eac h y ∈ Y w e ma y c ho ose m ( y ) ∈ N suc h that (7.1) a + α ≥ h ( T , π − 1 ( y )) + α ≥ 1 m ( y ) log r m ( y ) ( d X , T , ǫ, π − 1 ( y )) . Let E y b e an ( m ( y ) , ǫ )-spanning set of π − 1 ( y ) w.r.t. T with the minim um cardinalit y . Then U y . = S z ∈ E y B m ( y ) ( z , 2 ǫ ) is an op en neigh bo rho o d of π − 1 ( y ), where B m ( y ) ( z , 2 ǫ ) denotes the op en ball in X with ce nte r z a nd radius 2 ǫ (in the se nse of ( d X ) m ( y ) - metric). Since the map π − 1 : Y → 2 X , y 7→ π − 1 ( y ) is upp er semi-con tin uous, there exists an op en neigh b orho o d W y of y for whic h π − 1 ( W y ) ⊆ U y . By the compactness W en Hu ang, Xiangdong Y e and Guoh ua Zhang 25 of Y there exist { y 1 , · · · , y k } ⊆ Y suc h that W . = { W y 1 , · · · , W y k } forms an op en co v er of Y . Let δ > 0 b e a Leb esgue n um b er of W a nd M = max { m ( y 1 ) , · · · , m ( y k ) } . Let π ( E ) n b e an y ( n, δ )-spanning set for π ( E ) w.r.t. S with the minim um car- dinalit y . F or eac h y ∈ π ( E ) n and 0 ≤ j < n , pic k c j ( y ) ∈ { y 1 , · · · , y k } with B ( S j ( y ) , δ ) ⊆ W c j ( y ) , where B ( S j ( y ) , δ ) denotes the o pen ball in Y with cen ter S j ( y ) and radius δ . Now define recursiv ely t 0 ( y ) = 0 and t s +1 ( y ) = t s ( y ) + m ( c t s ( y ) ( y )) ( s ∈ Z + ) until one gets a t q +1 ( y ) ≥ n ; set q ( y ) = q ≤ t q ( y ). F or an y y ∈ π ( E ) n and z 0 ∈ E c t 0 ( y ) ( y ) , z 1 ∈ E c t 1 ( y ) ( y ) , · · · , z q ( y ) ∈ E c t q ( y ) ( y ) ( y ) w e define V ( y ; z 0 , z 1 , · · · , z q ( y ) ) = { u ∈ X : ( d X ) m ( c t s ( y ) ( y )) ( T t s ( y ) ( u ) , z s ) < 2 ǫ, 0 ≤ s ≤ q ( y ) } . Ob viously , f o r eac h y ∈ π ( E ) n , the num b er o f p ermissible tuples ( z 0 , z 1 , · · · , z q ( y ) ) is (7.2) N y = q ( y ) Y s =0 r m ( c t s ( y ) ( y )) ( d X , T , ǫ, π − 1 ( c t s ( y ) ( y ))) . Then w e hav e (using ( 7 .1) and (7.2)) (7.3) N y ≤ q ( y ) Y s =0 e ( a + α ) m ( c t s ( y ) ( y )) = e ( a + α )( t q ( y ) ( y )+ m ( c t q ( y ) ( y ) ( y ))) ≤ e ( a + α )( n + M ) . Note that if F is a n ( n, 4 ǫ )-separated subset of E w.r.t. T then, for each permissible tuple ( z 0 , z 1 , · · · , z q ( y ) ), V ( y ; z 0 , z 1 , · · · , z q ( y ) ) ∩ F has at most one elemen t, and [ y ∈ π ( E ) n    [ z s ∈ E c t i ( y ) ( y ) , 0 ≤ s ≤ q ( y ) V ( y ; z 0 , z 1 , · · · , z q ( y ) )    ⊇ E . Th us com bining (7.3) w e ha v e s n ( d X , T , 4 ǫ, E ) ≤ X y ∈ π ( E ) n N y ≤ r n ( d Y , S, δ, π ( E )) e ( a + α )( n + M ) . Letting n → + ∞ one has s ( d X , T , 4 ǫ, E ) ≤ r ( d Y , S, δ, π ( E )) + a + α ≤ h ( S, π ( E )) + a + α . Since ǫ > 0 and α > 0 are a rbitrary , w e obtain h ( T , E ) ≤ h ( S, π ( E )) + a . This finishes the pro of.  As a direct consequence of Theorem 7.3, w e hav e the follow ing prop osition. Prop osition 7.4. L et π : ( X , T ) → ( Y , S ) b e a factor map b etwe en GTDSs. If ( Y , S ) has finite entr op y, then sup E ∈ 2 X ( h ( T , E ) − h ( S, π ( E )) = h top ( T , X | π ) . 26 lo weri ng top olog ical entrop y ov er subsets Pr o of. On one hand we know sup E ∈ 2 X ( h ( T , E ) − h ( S, π ( E )) ≤ h top ( T , X | π ) b y The- orem 7.3. On the other hand, w e hav e sup E ∈ 2 X ( h ( T , E ) − h ( S, π ( E )) ≥ sup y ∈ Y ( h ( T , π − 1 ( y )) − h ( S, { y } )) = sup y ∈ Y h ( T , π − 1 ( y )) = h top ( T , X | π ) . This completes the pro of.  Let π : ( X , T ) → ( Y , S ) b e a factor map b etw een GTDSs. W e call that π is a prin- cip a l factor map (or ( X , T ) is a princ i p al extension of ( Y , S )) if h µ ( T , X ) = h π µ ( S, Y ) for eac h µ ∈ M ( X , T ). This w as intro duced a nd studied firstly b y Ledrappier [18]. The following result is a direct consequence of Lemma 7.2 and Theorem 7.3. Corollary 7.5. L et π : ( X , T ) → ( Y , S ) b e a princ i p al factor m a p b etwe ens GTDS s. If ( Y , S ) h a s finite entr opy then h top ( T , X | π ) = 0 a nd h ( T , K ) = h ( S, π ( K )) for al l K ∈ 2 X . A characterization o f asymptotical h -expansiv eness is obtained recen tly b y Bo yle and D o wnaro wicz as the fo llowing [5 , Theorem 8.6]: Lemma 7.6. A T D S ( X , T ) is asymptotic al ly h -exp ansiv e i ff it admits a princip al extension to a symb olic TDS. Then question 1.4 is answe red as follows : Theorem 7.7. A TDS ( X , T ) is asymptotic al ly h -exp ansive iff it is HUL . Pr o of. First, eac h HUL TDS is asymptotically h -expansiv e b y Theorem 6.4. No w assume that TDS ( X, T ) is asymptotically h -expansiv e and by Lemma 7.6 let π : ( X ′ , T ′ ) → ( X , T ) be a principal f a ctor map with ( X ′ , T ′ ) a s ym b olic TDS. Then h top ( T , X ) ≤ h top ( T ′ , X ′ ) < + ∞ (as ( X ′ , T ′ ) is a sym b olic TDS), and so for eac h E ∈ 2 X ′ , w e ha v e h ( T ′ , E ) = h ( T , π ( E )) b y Coro llary 7.5. Giv en E ∈ 2 X . Since ( X ′ , T ′ ) is an expansiv e TDS, then using Theorem 5 .4 w e ha v e that for eac h 0 ≤ h ≤ h ( T , E ) = h ( T ′ , π − 1 ( E )) there exists a coun table compact subset X ′ h ⊆ π − 1 ( E ) with at most a limit point in X ′ suc h that h ( T ′ , X ′ h ) = h . No w set X h = π ( X ′ h ) ⊆ E . So X h ⊆ X is a coun table compact subset with at most a limit p oin t in X and h ( T , X h ) = h ( T ′ , X ′ h ) = h . That is, TDS ( X , T ) is HUL .  Moreo v er, combining with Corollary 7.5 and Theorem 7.7 w e claim that principal extension preserv es the pro perties of low ering, hereditary low ering and HUL . Prop osition 7.8. L et π : ( X ′ , T ′ ) → ( X , T ) b e a princip a l factor map b etwe en TDSs. If ( X , T ) has finite entr opy then (1) ( X ′ , T ′ ) is asymptotic al ly h -exp ansive iff so i s ( X , T ) . (2) ( X ′ , T ′ ) is lower able (r e s p . h er e ditarily lowe r able, HUL ) iff s o is ( X , T ) . Pr o of. (1) is only a sp ecial case of Ledrappier’s result ab out principal extensions [18, Thorem 3]. (2) follows directly f rom Coro llary 7 .5, Theorem 7.7 and ( 1 ).  W en Hu ang, Xiangdong Y e and Guoh ua Zhang 27 It is not hard to construct examples with infinite en tropy whic h ar e hereditarily lo w erable. Th us, there are TD Ss whic h are hereditarily lo w erable but not HUL . In fact, an example with the same prop erty which ha s finite entrop y exists. Example 7.9. T her e exis ts a her e ditarily lower able TDS ( X, T ) wi th finite entr opy which is not HUL . The detaile d c ons truction is given as fo l lows: T ak e a countable copies of the full shift ov er { 0 , 1 } Z and em b ed them in to B n with { B n } n ∈ N a sequence of disjoin t compact ba lls in R 2 suc h that (0 , 0) / ∈ B n → { (0 , 0) } (in the sense of Hausdorff metric). Let ( X, T ) b e the TD S of the union of { (0 , 0) } with these copies, wh ere T is the shift if it is restricted on each copy and (0 , 0) 7→ (0 , 0). Then h top ( T , X ) = log 2 (using the classical v ariational principle). F or eac h cop y we ma y take C n ⊆ B n with entrop y a n < log 2 suc h that lim n → + ∞ a n = lo g 2 (using Theorem 4.4). Then h ( T , S n ∈ N C n ∪ { (0 , 0) } ) = log 2. Whereas, b y definition it is not ha rd to see that an y coun table compact subset of X with a unique limit p oin t (0 , 0) m ust ha v e zero e ntrop y , whic h implies that eac h countable compact subset of S n ∈ N C n ∪ { (0 , 0) } with a unique limit p oin t has en tropy smaller strictly than log 2. Thus ( X , T ) is not HUL . No w w e claim tha t ( X , T ) is here ditarily lo werable. F or eac h n ∈ N , w e take x n ∈ B n . Then it is no t ha r d to see that h ( T , { x n } n ∈ N ∪ { (0 , 0) } ) = 0. Let K ∈ 2 X and 0 ≤ h ≤ h ( T , K ). F or eac h n ∈ N , set K n = K ∩ B n . Since B n is HUL (see Theorem 5.4), we ma y tak e K h n ∈ 2 K n with h ( T , K h n ) = min { h, h ( T , K n ) } . Using Lemma 6.2 w e hav e sup n ∈ N h ( T , K n ) = max  sup n ∈ N h ( T , K n ∪ { x n } ) , h ( T , { x n } n ∈ N ∪ { (0 , 0) } )  = h T , [ n ∈ N ( K n ∪ { x n } ) ∪ { (0 , 0) } ! ≥ h ( T , K ) ≥ h. This implies sup n ∈ N h ( T , K h n ) = h . Let K h = S n ∈ N K h n ∪ { ( 0 , 0) } ∈ 2 K . Then using Lemma 6.2 a gain one has h ( T , K h ) = sup n ∈ N h ( T , K h n ) = h . This means that K is lo w erable, and so ( X , T ) is a hereditarily low erable TDS. This ends the example. It is not difficult to show that the ab o v e example (by a small mo dification) has a sym b o lic extension with the same en trop y , whic h is not a principal one (see [5] for other examples of the same type). Th us it is an interes ting question if each system ha ving a sym b olic extension is hereditarily lo w erable. 8. Appendix In this App endix w e wan t to explain that our main results hold for G TD Ss. Note that w e can also introduce the lo w erable, hereditarily low erable, HUL a nd asymptotically h -expansiv e pro perties for a GTDS. Let ( X, T ) b e a G TDS. If T is surjectiv e, we can use the standard natural extension as follows: Assume that d is a metric on X . W e say ( X T , S ) is the na tur al extension of ( X , T ), if X T = { ( x 1 , x 2 , · · · ) : T ( x i +1 ) = x i , x i ∈ X , i ∈ N } , which is a sub-space of the 28 lo weri ng top olog ical entrop y ov er subsets pro duct space Q ∞ i =1 X with the compatible metric d T defined b y d T (( x 1 , x 2 , · · · ) , ( y 1 , y 2 , · · · )) = ∞ X i =1 d ( x i , y i ) 2 i . Moreo v er, S : X T → X T is the shift homeomorphism, i.e. S ( x 1 , x 2 , x 3 , · · · ) = ( T ( x 1 ) , x 1 , x 2 , · · · ). The following observ a tion is easy . Theorem 8.1. L et ( X T , S ) b e the natur al extension of ( X, T ) with T surje ctive. Then ( X T , S ) is low er able (r esp. her e d itarily lowe r able, H UL , asymp totic al l y h - exp a nsive) iff so is ( X , T ) . Pr o of. Let π 1 : X T → X b e the pro jection to the first co ordinate. Observ e that diam( S n π − 1 1 ( x )) → 0 for each x ∈ X . This implies that h ( S, π − 1 1 ( x )) = 0 fo r eac h x ∈ X , and hence h top ( S, X T | π 1 ) = sup x ∈ X h ( S, π − 1 1 ( x )) = 0 . By Theorem 7 .3, ( X T , S ) is low erable (resp. hereditarily low erable, HUL ) iff so is ( X , T ). Since h ( S , π − 1 1 ( x )) = 0 for each x ∈ X , π 1 is a principal extension b y L emma 7.2. No w as a principal extension preserv es the prop ert y of asymptotical h - expans iv eness (see [1 8, Theorem 3]), ( X T , S ) is asymptotically h -expansiv e iff so is ( X, T ).  If T is not surjectiv e, w e will construct a surjectiv e system ( X ′ , T ′ ) such t ha t the dynamical prop erties of ( X , T ) and ( X ′ , T ′ ) a re ’very close’ as follows: Let X ′ = X × { 0 } ∪ X × { 1 n : n ∈ N } . Moreo v er, put T ′ ( x, 0) = ( x, 0) , T ′ ( x, 1 n +1 ) = ( x, 1 n ) a nd T ′ ( x, 1) = ( T x, 1) fo r n ∈ N and x ∈ X . It is not hard to chec k tha t ( X ′ , T ′ ) is low erable (resp. hereditarily low erable, HUL , asymptotically h -expansiv e) iff so is ( X , T ). Collecting terms, one has Theorem 8.2. L et ( X , T ) b e a GTDS. Then ( X , T ) is HUL iff it is as ymptotic al ly h -exp a nsive, and if ( X , T ) has finite entr opy then it is lower able. Reference s [1] T. 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Y e a nd G. H. Z hang, Entr opy p oints and applic ations , T ra ns . Amer. Math. So c. 359 (2007), 6167- 6186. Dep ar tment of Ma thema tics, U niversity of Science and Technology of China, Hefei, Anhui, 230026, P.R. China E-mail addr ess : wenh@ mail.ust c.edu.cn, yexd@ustc .edu.cn School of Ma thema tical Sciences, Fudan University, Shanghai 20043 3, China E-mail addr ess : zhang gh@fudan .edu.cn