Adjoint entropy vs Topological entropy

Adjoin t en trop y vs T op ological en trop y Anna Giordano Bruno anna.giord anobruno@uniud.i t Dipartimento di Matematica e Informatica, Universit` a di Udine, Via d elle Scienze, 206 - 33100 Udine Dedicated to the sixtieth birthda y of Dikran Dikranjan Abstract Recently the adjoint algebraic entrop y of endomorphisms of abelia n groups w as intro duced and studied in [7]. W e generalize the notion of a djo int entropy to co n tin uous endomorphisms of to pologica l ab elian groups. Indeed, the adjoint algebraic en tropy is defined using the family of all finite- ind ex subg r oups, while we take only the subfamily o f all op en finite-index subgroups to define the topo logical adjoint en tropy . This allows us to compare the (topo logical) adjoin t entrop y with the known top ological entropy of contin uous endomorphisms of compact ab e lia n gr o ups. In particular , the top ological adjoint entrop y and the top ological entropy co incide on contin uous endomo rphisms of tota lly disco nnec t ed compact ab elian gro ups. Mo reo ver, we pr o ve tw o Bridge Theorems b et ween the to pologica l adjoint ent ro p y and the algebraic entropy using r espectively the Pontry agin duality and the precompact dualit y . 1 In tro duction In the same pap er [1] where they in tro duced and studied the top ological en tropy h top for con tinuous maps of compact spa ces (see Section 5 for the precise definition), Adler , K onheim a nd McAndrew defined the algebraic ent ro p y of endomorphisms of ab elian groups. The notion of alg ebraic en tropy was studied later by W eiss [19], who gav e its basic prop erties a nd its relation with the top ological en tropy . The a lgebraic ent ro p y of an e nd omo rphism φ of an ab elian g r oup G measures to what exten t φ mov es the finite subgroups F of G . More pre c isely , according to [1, 19], for a p ositiv e integer n , let T n ( φ, F ) = F + φ ( F ) + . . . + φ n − 1 ( F ) be the n -th φ -t r aje ctory of F . The algebr aic entro py of φ with r esp e ct to F is H ( φ, F ) = lim n →∞ log | T n ( φ, F ) | n , and the algebr aic entro py of φ : G → G is ent ( φ ) = sup { H ( φ, F ) : F is a finite subgr oup of G } . Clearly ent( φ ) = ent ( φ ↾ t ( G ) ), wher e t ( G ) denotes the tors ion part of G ; so the natural co n tex t for the alg ebraic ent ro p y is that of endomorphis m s of torsion ab elian gr oups. Many pap ers in the la st years were dev oted to the study o f v ario us asp ects of the algebraic entropy , a nd mainly [8], where its fundamen tal prop erties were prov ed. In analogy to the alg ebraic ent ro p y , in [7] the adjoint algebraic ent ro p y o f endomorphisms φ o f a belian groups G was intro duced “ replacing” the family o f all finite subgroups of G with the family C ( G ) of a ll finite-index subgro ups of G . Indeed, for N ∈ C ( G ), and a p ositive integer n , let B n ( φ, N ) = N ∩ φ − 1 ( N ) ∩ . . . ∩ φ − n +1 ( N ); 1 the n -th φ -c otr aje ctory of N is C n ( φ, N ) = G B n ( φ, N ) . Each C n ( φ, N ) is finite, a s each B n ( φ, N ) ∈ C ( G ), becaus e the family C ( G ) is stable under inv erse images with resp ect to φ and under finite intersections. The adjoi nt algebr aic entro py of φ with r esp e ct to N is H ⋆ ( φ, N ) = lim n →∞ log | C n ( φ, N ) | n . (1.1) This limit exists and it is finite; in fact, as shown in [7, P ropositio n 2.3], the sequence α n =     C n +1 ( φ, N ) C n ( φ, N )     =     B n ( φ, N ) B n +1 ( φ, N )     is stationary; (1.2) more precisely , there exists a natural n umber α such that α n = α for all n larg e enough. The adjoint algebr aic entr opy of φ : G → G is ent ⋆ ( φ ) = sup { H ⋆ ( φ, N ) : N ∈ C ( G ) } . While the algebraic entrop y has v alues in log N + ∪ {∞} , the adjoint algebraic entrop y takes v alues only in { 0 , ∞} a s shown by [7, Theorem 7 .6]. This pa rticular “binary b ehavior” of the v alues o f the adjoint algebraic ent ro p y se e ms to be caused b y the fact that the family of finite-index subgr oups can be very la rge. So in this pap er w e ta k e only a part of it using the top ology in the following w ay . F or a topo logical ab elian gro up ( G, τ ), consider the subfamily C τ ( G ) = { N ∈ C ( G ) : N τ -o pen } of C ( G ) consisting of a ll τ -op en finite-index subgro ups of G . Definition 1.1. F or a t op olo gic al ab elian gr oup ( G, τ ) and a c ontinuous endomorphism φ : ( G, τ ) → ( G, τ ) , the top o logical a dj oint en tropy of φ with re sp e ct t o τ is ent ⋆ τ ( φ ) = sup { H ⋆ ( φ, N ) : N ∈ C τ ( G ) } . (1.3) Roughly sp eaking, ent ⋆ τ is a v aria n t of e nt ⋆ but taken only with r espect to some finite-index subgro ups, namely , the τ -op en ones. Clear ly , for ( G, τ ) a topolo gical abelian gr oup and φ : ( G, τ ) → ( G, τ ) a contin uous endomorphism, ent ⋆ ( φ ) ≥ ent ⋆ τ ( φ ). F or the dis crete topolo gy δ G of G , we have ent ⋆ δ G ( φ ) = ent ⋆ ( φ ), so the notion of top ological adjoint entropy extends that o f adjoint a lgebraic entropy , and this provides a first motiv ation for this pa per. In Section 3 w e study the basic prop erties of the top ological adjoin t entrop y , with resp ect to the t ypica l prop erties of the known entropies. Note that we defined the topo logical adjoin t en tro p y for a c ontinuous endomo rphism φ : ( G, τ ) → ( G, τ ) of a top o logical ab elian group ( G, τ ). F rom a categor ical point of v ie w this seems to be the rig h t setting, also as C τ ( G ) is stable under inv erse images with resp ect to φ just b ecause φ is co n tin uous. O n the other hand, the definition formally mak es sense for not nece ssarily contin uous endomor phisms, and furthermore the basic prop erties proved in Section 3 hold true even re moving the contin uit y of the endomor phisms. So it may b e desirable to study also this mor e gener al situation. Recall that a topolo gical ab elian group ( G, τ ) is t o tal ly b ounde d if for every op en neighbor hoo d U of 0 in ( G, τ ) there exists a finite subset F o f G such that U + F = G . Mor eo ver, ( G, τ ) is pr e c omp act if it is Hausdorff and totally b ounded. Since the co mp letion ^ ( G, τ ) of a prec ompact ab elian gr oup ( G, τ ) is compact, the preco mpact ab elian groups are precis e ly the subgroups of compa c t ab elian gro ups. Moreov er, the group top ology τ is said to be line ar if it has a base of the neighbor hoo ds of 0 co ns isting of op en subgroups. In Section 4 we consider other prop erties of the to pologica l adjoint entrop y , giving top ological and alg ebraic reductions for its computation. In pa rticular, we see that in order to co mpu te the top ological adjoint en tro p y of contin uous endomor phisms of top ological ab elian groups, it suffices to consider pr ecompact linear topolo gies (see Remark 4.12). In Section 5, which co n tains the main results of this pap er, we study the connections of the top ological adjoint entropy with the top ological entrop y and the algebraic ent ro p y . Note that the adjoint a lgebraic en tro p y is closely related to the algebraic e ntropy through the Pontry ag in duality (see Section 2 for basic definitions a nd prop erties of the Pon tryagin duality , see also [12, 15]). Indeed, [7, Theorem 5.3], which is a so-called Bridge Theorem, shows tha t : 2 Theorem 1.2 (Br idge Theorem ent ⋆ -ent) . The adjoint ent r opy of an endomorphism φ of an ab elian gr oup G is the same as the algebr aic entr opy of the dual endomorph ism b φ of φ of the (c omp act) Pontryagin dual b G of G . Moreov er , W eiss prov ed the Bridge Theorem [1 9 , Theorem 2 .1] relating the to p ological entropy a nd the algebraic entrop y through the Pon tryagin duality: Theorem 1 .3 (Bridg e Theorem h top -ent) . The top olo gic al entr opy of a c ontinuous endomorp hism φ of a total ly disc onn e cte d c omp act ab elian gr oup ( G, τ ) is the same as t h e algeb r aic entr opy of the dual endomorp hism b φ of φ of the (t o rsion and discr et e) Pontryagin dual \ ( G, τ ) of ( G, τ ) . First of all we s ee tha t the topo lo gical a djoin t entrop y app eared in s ome non-ex plicit way in the pro of of this clas sical theor em of W eiss. More precisely , The o rem 5.3 shows that t h e top olo gic al adj oint entr opy c oincides with the top olo gic al entr opy for co n tin uous endomorphisms of totally disco nnected compact ab elian groups (in particula r, the top ological adjoint entrop y tak es ev ery v alue in lo g N + ∪ {∞} , a more natural behavior with r e spect to that o f the adjoint a lgebraic entrop y). Therefore, the top ological adjoint entrop y comes as an alternative form of topo logical en tropy , with the adv antage tha t it is defined for every co n tin uous endomorphism o f every ab elian top ological group, and not only in the compact cas e . So this provides a second motiv ation for intro ducing the a djo int top ological en tropy . Here comes a third motiv ation for the top ological adjoint entropy . Indeed, the following B r idge Theo rem 1.4, relating the topolog ical adjoint entrop y and the algebraic entropy throug h the P ontryagin duality , extends W eiss’ Bridge Theo rem 1 .3 from totally disconnected compac t ab elian groups to arbitrar y co mpact a belian groups. On the other hand, this is not p ossible using the topo logical entrop y , so in some sens e the adjoint top ological e ntropy may be interpreted as the true “dual entropy” of the algebra ic entrop y . Theorem 1.4 (Br idge Theo r em ent ⋆ τ -ent) . The top olo gic al adjoint entr opy of a c ontinu ous endomorp hism φ of a c omp act ab elian gr oup ( G, τ ) is the same as t h e algebr aic entr opy of the dual endomorphism b φ of φ of t he (discr ete) Pontryagin dual \ ( G, τ ) of ( G, τ ) . In analogy to the Bridge Theorem 1.2 b et ween the adjoint algebr aic entropy a nd the a lgebraic en tro py , we give also ano ther Br idge The o rem, making use of the precompact duality (see Section 2 for the basic definitions and prop erties of the precompact duality , see also [14, 17]): Theorem 1.5 (Br idge Theo r em ent ⋆ τ -ent) . The top olo gic al adjoint entr opy of a c ontinu ous endomorp hism φ of a pr e c omp act ab elian gr oup ( G, τ ) is the same as t he algebr aic entr opy of the dual endomorphism φ † of φ of the pr e c omp act dual ( G, τ ) † of ( G, τ ) . The main examples in the context of ent ro p y are the Ber noulli shifts; fo r K a n ab elian group, (a) the right Bernoul li shift β K and the left Bernoul li s h ift K β of the gro up K N are de fined resp ectiv ely b y β K ( x 0 , x 1 , x 2 , . . . ) = (0 , x 0 , x 1 , . . . ) a nd K β ( x 0 , x 1 , x 2 , . . . ) = ( x 1 , x 2 , x 3 , . . . ); (b) the two-side d Bernoul li shift β K of the gro up K Z is defined by β K (( x n ) n ∈ Z ) = ( x n − 1 ) n ∈ Z , for ( x n ) n ∈ Z ∈ K Z . Since K ( N ) is bo th β K -inv a rian t and K β -inv a rian t, and K ( Z ) is β K -inv a rian t, let β ⊕ K = β K ↾ K ( N ) , K β ⊕ = K β ↾ K ( N ) and β ⊕ K = β K ↾ K ( Z ) . The a djoin t a lg ebraic entrop y ta k es v alue infinit y on the Ber noulli shifts, that is , if K is a non-trivial abelia n group, then ent ⋆ ( β ⊕ K ) = ent ⋆ ( K β ⊕ ) = ent ⋆ ( β ⊕ K ) = ∞ . These v alues were ca lculated in [7, P ropos itio n 6.1] applying [1 1, Coro llary 6.5] and the Pon tryagin duality (see E xample 6.1); we giv e here a direct computation in Pr oposition 6.2. This result w as one o f the main steps in proving the ab o ve mentioned [7, Theo rem 7.6] showing that the adjoint algebraic en tropy takes v alues only in { 0 , ∞} . 3 Akno wledgemen ts I w ould like to thank Profess or Dikranjan for his very useful commen ts and sug g estions. Notation and T erminology W e denote b y Z , N , N + , Q and R r espectively the integers, the natura l num ber s, the p o sitiv e integers, the rationals and the re als. F or m ∈ N + , w e use Z ( m ) for the finite cyclic gr oup of order m . Moreov er , w e consider T = R / Z endowed with its compact top ology . Let G b e an a belian gro up . F or a s et X we denote by G ( X ) the direct sum L X G of | X | many copies of G . Moreov er , End( G ) is the ring of all endo morphisms o f G . W e denote by 0 G and id G resp ectiv ely the endomorphism of G which is iden tically 0 and the identit y endomor phism of G . F or a top ological abelia n g roup ( G, τ ), if X is a s ubset o f G , then X τ denotes the closure of X in ( G, τ ); if there is no po ssibilit y of confusion we wr ite simply X . Moreov er, if H is a subgr o up o f G , we denote b y τ q the quotient top ology of τ on G/H . If ( G, τ ) is Haus dorff, then ^ ( G, τ ) is the completion of ( G, τ ) and for φ : ( G, τ ) → ( G, τ ) a co n tinuous endomorphism, e φ : ^ ( G, τ ) → ^ ( G, τ ) is the (unique) contin uous extension o f φ to ^ ( G, τ ) . F ur t hermo re, c ( G ) deno t es the co nnected comp onen t of G . 2 Bac kground on dualities Let ( G, τ ) be an a belian group. The dual g roup ( G, τ ) ′ of ( G, τ ) is the gro up of a ll contin uo us homo morphisms ( G, τ ) → T , endow ed with the discrete top ology . If φ : ( G, τ ) → ( G, τ ) is a contin uous endomor phism, its dual endomorphism φ ′ : ( G, τ ) ′ → ( G, τ ) ′ is defined by φ ′ ( χ ) = χ ◦ φ for every χ ∈ ( G, τ ) ′ . F or A ⊆ G and B ⊆ ( G, τ ) ′ , the annihi lator A ⊥ of A in ( G, τ ) ′ and the annihilator B ⊤ of B in G are resp ectiv ely A ⊥ = { χ ∈ ( G, τ ) ′ : χ ( A ) = 0 } and B ⊤ = { g ∈ G : χ ( g ) = 0 , ∀ χ ∈ B } . Moreov er , one can consider the map ω G : ( G, τ ) → ( G, τ ) ′′ defined by ω G ( g )( χ ) = χ ( g ) for every g ∈ G and χ ∈ ( G, τ ) ′ . F or a lo cally compact a belian group ( G, τ ) the Pon tryagin dual \ ( G, τ ) of ( G, τ ) is ( G, τ ) ′ endow ed with the compact-op en top ology [15], denoted here by b τ . The Pontry ag in dual of a lo cally compact ab elian gr oup is lo cally compact a s w ell, and the Pontry agin dual of a (discr e te) ab elian gro up is alw ays compact [12, 1 5 ]. The map ω G : ( G, τ ) → \ \ ( G, τ ) is a top ological isomorphism. Mor e over, for a contin uos endomor phism φ : ( G, τ ) → ( G, τ ), its dual endomorphis m φ ′ : \ ( G, τ ) → \ ( G, τ ) is contin uo us, and it is usually denoted by b φ : \ ( G, τ ) → \ ( G, τ ) . F or bas ic pro p erties concerning the Pon tryagin duality see [9, 12]. W e pas s now to the precompact duality , studied mainly in [14, 17]. Assume that ( G, τ ) is precompact. Since ( G, τ ) is dense in its co mpact completion K = ^ ( G, τ ) , and since the contin uous c hara cters of ( G, τ ) are uniformly contin uous, they can b e extended to K , and it fo llo ws that ( G, τ ) ′ = K ′ . The pr e c omp act dual ( G, τ ) † of a pr e compact a belian gr oup ( G, τ ) is ( G, τ ) ′ endow ed with the weak top ology τ † generated by all elements of G co nsidered as contin uous characters; this is the top ology of ( G, τ ) ′ inherited from the pro duct top ology of T G , since ( G, τ ) ′ can b e considered as a subgro up of T G . Then ( G, τ ) † is a precompact ab elian group. As prov ed in [17, Theor em 1], the map ω G : ( G, τ ) → ( G, τ ) †† is a top ological isomorphism. The same r esult follo ws from [14, P ropositio ns 2 .8, 3.9 and The o rem 3 .11]. Moreover, if φ : ( G, τ ) → ( G, τ ) is a contin uous endomorphism, then the dual endomor ph ism φ ′ : ( G, τ ) † → ( G, τ ) † is contin uous [17], and it is usually denoted by φ † : ( G, τ ) † → ( G, τ ) † . 4 The precompact dualit y has the sa me basic prop erties of the Pon tryagin dualit y . Indeed, also in this case the annihilator s are closed subgr oups of ( G, τ ) † and ( G, τ ); furthermore, by [17, Rema rk 10 ], if ( G, τ ) is a precompact ab elian group and H a closed subgr oup of ( G, τ ), then: (a) ( G/H , τ q ) † is top ologically isomorphic to ( H ⊥ , τ † ↾ H ⊥ ); (b) ( H , τ ↾ H ) † is topo logically is omorphic to (( G, τ ) † /H ⊥ , ( τ † ) q ); (c) the map H 7→ H ⊥ defines an order-inverting bijection betw een the clo sed subgro ups of ( G, τ ) and the closed subgroups of ( G, τ ) † . Moreov er , it is po ssible to prov e the following pro perties, exactly as their counterparts for the P ontry ag in duality (see [1 2 ] for (a) and (b), and [7] for (c)). F act 2 . 1. (a) If F is a fin i te ab elian gr oup, then F † ∼ = F . (b) If H 1 , . . . , H n ar e sub gr oups of a pr e c omp act ab elian gr oup G , then ( P n i =1 H i ) ⊥ ∼ = top T n i =1 H ⊥ i and ( T n i =1 H i ) ⊥ ∼ = top P n i =1 H ⊥ i . (c) If G is a pr e c omp act ab elian gr oup, H a sub gr oup of G and φ : G → G a c ontinu o us endomorphi sm, then ( φ − n ( H )) ⊥ = ( φ † ) n ( H ⊥ ) for every n ∈ N . 3 Basic prop erties of the top ological adjoin t en trop y In this section we show the basic pr operties of the top ological adjoint ent ro p y , with r e s pect to the usual basic prop erties poss essed by the known entropies, as in particular the a lgebraic entropy and the to p ological ent ro p y . These pro perties w ere discuss ed in [7, Section 8] for the a djo int algebra ic en tropy . Since o ur setting inv o lving top ology is more general, they require a new verification, even if formulas from [7] are a pplied in the pro ofs. W e start giving the following ea sy obser v a tion ab out the monotonicity of H ⋆ ( φ, − ) with resp ect to sub- groups. Lemma 3 .1. L et G b e an ab elian gr oup, φ ∈ End( G ) and N , M ∈ C ( G ) . If N ⊆ M , then B n ( φ, N ) ⊆ B n ( φ, M ) and so | C n ( φ, N ) | ≥ | C n ( φ, M ) | for every n ∈ N + . Ther efor e, H ⋆ ( φ, N ) ≥ H ⋆ ( φ, M ) . W e have also the following monotonicity with r espect to the top ology . Lemma 3.2. L et G b e an ab elian gr oup and τ , τ ′ gr oup top olo gies on G . If τ ≤ τ ′ on G , then ent ⋆ τ ( φ ) ≤ ent ⋆ τ ′ ( φ ) . Example 3.3 . It easily follows from the definition that en t ⋆ τ (0 G ) = ent ⋆ τ ( id G ) = 0 for an y top ological ab elian group ( G, τ ). The first prop ert y is the inv ariance under conjuga t ion. Lemma 3. 4. L et ( G, τ ) b e a top olo gic al ab elian gr oup and φ : ( G, τ ) → ( G, τ ) a c ontinu o us endomorphism. If ( H , σ ) is another top olo gic al ab elian gr oup and ξ : ( G, τ ) → ( H , σ ) a top olo gic al isomorphism, then ent ⋆ σ ( ξ ◦ φ ◦ ξ − 1 ) = ent ⋆ τ ( φ ) . Pr o of. Let N ∈ C σ ( H ) and c a ll θ = ξ ◦ φ ◦ ξ − 1 . Since ξ is a top ological isomorphism, ξ − 1 ( N ) ∈ C τ ( G ). In the pro of of [7 , Lemma 4 .3 ], it is shown that H ⋆ ( θ, N ) = H ⋆ ( φ, ξ − 1 ( N )) for every N ∈ C σ ( H ), a nd hence ent ⋆ σ ( θ ) = ent ⋆ τ ( φ ). The second prop erty is the so- called loga rithmic law. Lemma 3. 5. L et ( G, τ ) b e a top olo gic al ab elian gr oup and φ : ( G, τ ) → ( G, τ ) a c ontinu o us endomorphism. Then for every k ∈ N + , ent ⋆ τ ( φ k ) = k · ent ⋆ τ ( φ ) . 5 Pr o of. F or N ∈ C τ ( G ), fixed k ∈ N + , for every n ∈ N + we ha ve C nk ( φ, N ) = C n ( φ k , B k ( φ, N )). Then, by the pro of o f [7, Lemma 4.4 ], k · H ⋆ ( φ, N ) = H ⋆ ( φ k , B k ( φ, N )) ≤ en t ⋆ τ ( φ k ) . Consequently , k · ent ⋆ τ ( φ ) ≤ en t ⋆ τ ( φ k ). Now we prov e the conv ers e ineq ua lit y . Indeed, by the pro of of [7, L e mm a 4 .4], for N ∈ C τ ( G ) and for k ∈ N + , ent ⋆ τ ( φ ) ≥ H ⋆ ( φ, N ) ≥ H ⋆ ( φ k , N ) k . This shows that k · ent ⋆ τ ( φ ) ≥ en t ⋆ τ ( φ k ), that concludes the pro of. The next lemma shows that a top ological automorphism has the same top ological a djo in t entropy as its inv e rse. Lemma 3.6. L et ( G, τ ) b e a top olo gic al ab elian gr oup and φ : ( G, τ ) → ( G, τ ) a top olo gic al automorphi sm. Then ent ⋆ τ ( φ ) = ent ⋆ τ ( φ − 1 ) . Pr o of. F or every n ∈ N + and every N ∈ C τ ( G ), we ha ve H ⋆ ( φ, N ) = H ⋆ ( φ − 1 , N ) by the pro of of [7, Lemma 4.5], and hence ent ⋆ τ ( φ ) = en t ⋆ τ ( φ − 1 ). The following co r ollary is a dir ect c o nsequence of the pr evious tw o results. Corollary 3.7. Le t ( G, τ ) b e a top olo gic al ab elian gr oup and φ : ( G , τ ) → ( G, τ ) a top olo gic al automorphism. Then ent ⋆ τ ( φ k ) = | k | · ent ⋆ τ ( φ ) for every k ∈ Z . The next prop ert y , a monotonicity la w for induced endomorphisms on quotien ts over in v aria n t subgroups, will b e often used in the sequel. Lemma 3. 8. L et ( G, τ ) b e a top olo gic al ab elian gr oup, φ : ( G, τ ) → ( G, τ ) a c ontinuous endomorphism and H a φ -invariant sub gr oup of G . Then en t ⋆ τ ( φ ) ≥ ent ⋆ τ q ( φ ) , wher e φ : ( G/H, τ q ) → ( G/H , τ q ) is the c ontinu o us endomorphi sm induc e d by φ . Pr o of. Let N /H ∈ C τ q ( G/H ); then N ∈ C τ ( G ). Since H ⊆ N a nd H is φ -inv ariant, H ⊆ φ − n ( N ) fo r every n ∈ N . Conse quen tly , H ⊆ B n ( φ, N ) for every n ∈ N + . Since φ − n ( N /H ) = φ − n ( N ) /H for ev er y n ∈ N , we hav e B n ( φ, N /H ) = B n ( φ, N ) /H for ev er y n ∈ N + . Therefor e, for every n ∈ N + , C n ( φ, N /H ) = ( G/H ) / ( B n ( φ, N ) /H ) ∼ = G/B n ( φ, N ) = C n ( φ, N ) . Hence, H ⋆ ( φ, N /H ) = H ⋆ ( φ, N ) , (3.1) and this prov es ent ⋆ ( φ ) ≤ en t ⋆ ( φ ). In gener al the top o logical adjoint entropy fails to b e monotone with resp ect to restrictions to inv a r ian t subgroups [7]. Nev ertheless , if we imp ose sufficien tly stronger conditions on the inv ariant subgroup, w e obtain more than the searched mono tonicit y in Lemma 3.9 and Prop osition 3.11 Lemma 3. 9. L et ( G, τ ) b e a top olo gic al ab elian gr oup, φ : ( G, τ ) → ( G, τ ) a c ontinuous endomorphism and H a φ -invaria nt sub gr oup of ( G, τ ) . If H ∈ C τ ( G ) , then e n t ⋆ τ ( φ ) = en t ⋆ τ ↾ H ( φ ↾ H ) . Pr o of. Let N ∈ C τ ↾ H ( H ). Since H ∈ C τ ( G ), it follows that N ∈ C τ ( G ) as w ell. By the pro of of [7, Lemma 4.9] this implies ent ⋆ τ ( φ ) ≥ en t ⋆ τ ↾ H ( φ ↾ H ). On the o ther hand, if N ∈ C τ ( G ), then N ∩ H ∈ C τ ↾ H ( H ), a nd B n ( φ, N ) ∩ H = B n ( φ ↾ H , N ∩ H ) for ev ery n ∈ N + . T her efore, H ⋆ ( φ, N ) = H ⋆ ( φ ↾ H , N ) and we can conclude that ent ⋆ τ ( φ ) ≤ en t ⋆ τ ↾ H ( φ ↾ H ). Hence, ent ⋆ τ ( φ ) ≤ en t ⋆ τ ↾ H ( φ ↾ H ). Let G b e an ab e lian gr oup and H a subgroup of H . Since there exists an injective homomorphism ι : H / N ∩ H → G/ N induced b y the inclusio n H ֒ → G , the map ξ : C ( G ) → C ( H ) defined by N 7→ N ∩ H is well-defined. Co ns ider now the g roup top ology τ on G . Then ξ restricts to ξ : C τ ( G ) → C τ ↾ H ( H ), and ι : H / N ∩ H → G/ N is contin uous with resp ect to the quo tien ts top ologies. Moreover, w e hav e the following Lemma 3 . 10. L et ( G, τ ) b e a top olo gic al ab elian gr oup and H a dense sub gr oup of ( G, τ ) . Then ξ : C τ ( G ) → C τ ↾ H ( H ) define d by N 7→ N ∩ H is a bije ction and its inverse is η : C τ ↾ H ( H ) → C τ ( G ) define d by M 7→ M . 6 Pr o of. Let N ∈ C τ ( G ). Then M = N ∩ H ∈ C τ ↾ H ( H ) and N = M in ( G, τ ); indeed, N ⊆ N ∩ H , becaus e H is dense in ( G, τ ), a nd so N = N ∩ H , since N is an open subgr oup and so clos ed. Prop osition 3.11 . L et ( G, τ ) b e a top olo gic al ab elian gr oup, φ : ( G, τ ) → ( G, τ ) a c ontinuous endomorphism and H a dense φ -invariant su b gr oup of ( G, τ ) . Then ent ⋆ τ ( φ ) = ent ⋆ τ ↾ H ( φ ↾ H ) . Pr o of. Let N ∈ C τ ( G ) and M = N ∩ H . By Lemma 3.10, the contin uous injective homomo rphism ι : H/ M → G/ N is also op en. Since H is de ns e in ( G, τ ), the image of H / M under ι is dense in G/ N , which is finite; hence H/ M ∼ = G/ N . Moreov er, H /B n ( φ ↾ H , M ) ∼ = G/B n ( φ, N ) for every n ∈ N + ; indeed, for every n ∈ N + , B n ( φ, N ) ∩ H = n − 1 \ i =0 φ − i ( N ) ∩ H = ∞ \ i =0 ( φ − i ( N ) ∩ H ) = n − 1 \ i =0 φ − i ( N ) = B n ( φ, M ) . Applying the definition, we ha ve H ⋆ τ ( φ, N ) = H ⋆ τ ( φ ↾ H , M ) and so ent ⋆ τ ( φ ) = en t ⋆ τ ↾ H ( φ ↾ H ). Corollary 3. 12. L et ( G, τ ) b e a Hausdorff ab elian gr oup, φ : ( G, τ ) → ( G, τ ) a c ontinuous endomorphism, and denote by e τ the top olo gy of t he c ompletion ^ ( G, τ ) of ( G, τ ) and by e φ : ^ ( G, τ ) → ^ ( G, τ ) the extension of φ . Then ent ⋆ τ ( φ ) = en t ⋆ e τ ( e φ ) . Now we v erify the additivity of the top ological adjoint entropy for the direct pro duct of tw o contin uous endomorphisms. Prop osition 3.13. L et ( G, τ ) b e a top olo gic al ab elian gr oup. If ( G, τ ) = ( G 1 , τ 1 ) × ( G 2 , τ 2 ) for some sub gr oups G 1 , G 2 of G with τ 1 = τ ↾ G 1 , τ 2 = τ ↾ G 2 , and φ = φ 1 × φ 2 : G → G for some c ontinuous φ 1 : ( G, τ 1 ) → ( G, τ 1 ) , φ 2 : ( G 2 , τ 2 ) → ( G 2 , τ 2 ) , t h en ent ⋆ τ ( φ ) = en t ⋆ τ 1 ( φ 1 ) + ent ⋆ τ 2 ( φ 2 ) . Pr o of. Let N ∈ C τ ( G ). Then N contains a s ubg roup o f the form N ′ = N 1 × N 2 , where N i = N ∩ G i ∈ C τ i ( G i ) for i = 1 , 2; in particular , N ′ ∈ C τ ( G ). F or e very n ∈ N + , we have | C n ( φ, N ) | ≤ | C n ( φ, N ′ ) | = | C n ( φ 1 , N 1 ) × C n ( φ 2 , N 2 ) | and so H ⋆ ( φ, N ) ≤ H ⋆ ( φ, N ′ ) = H ⋆ ( φ 1 , N 1 ) + H ⋆ ( φ 2 , N 2 ) . Now the thesis follows from the definition. The contin uit y for inv ers e limits fails in gener al, as shown in [7]. But it holds in the pa rticular case of contin uous endomorphisms φ of totally dis c onnected compact ab elian groups ( K , τ ). In fact, Theo rem 5.3 will show that in this c a se the top ological adjoint en tropy of φ coincides with the top ological ent ro p y of φ , and it is known that the to pological entrop y is contin uous for inv erse limits [1]. 4 T op ological adjoin t en trop y and functorial top ologies Let ( G, τ ) be a to pologica l abe lian group and φ : ( G, τ ) → ( G, τ ) a contin uous endomorphis m . Obviously , ent ⋆ τ ( φ ) ≤ en t ⋆ ( φ ) since the suprem um in the definition of the top ological adjoint entrop y (see (1.3)) is tak en ov er the subfamily C τ ( G ) of C ( G ) consisting only of the τ - open finite-index subg roups of G . F or an ab elian group G , the pr ofinite top olo gy γ G of G ha s C ( G ) as a base of the neighborho o ds of 0. So it is worth noting that C ( G ) = C γ G ( G ), and the next pro perties easily follow: Lemma 4. 1. L et ( G, τ ) b e a top olo gic al ab elian gr oup and φ : ( G, τ ) → ( G, τ ) a c ontinu o us endomorphism. Then: (a) ent ⋆ ( φ ) = en t ⋆ γ G ( φ ) ; (b) if τ ≥ γ G , t h en ent ⋆ τ ( φ ) = en t ⋆ γ G ( φ ) = en t ⋆ ( φ ) . 7 W e consider now tw o mo difications of a gro up top ology , which are functors T opAb → T opAb , where T opAb is the categor y of all top ological abe lia n gr oups and their contin uous homomo rphisms. If ( G, τ ) is a top ological abe lia n gr oup, the Bohr mo dific ation of τ is the topo logy τ + = sup { τ ′ : τ ′ ≤ τ , τ ′ totally b o unded } ; this topo logy is the fines t totally b ounded group top ology on G co arser than τ . Actually , τ + = inf { τ , P G } , where P G is the Bohr top olo gy , tha t is, the group top ology of G generated by all characters of G . Note that δ + G = γ G , where δ G denotes the discrete top ology of G . Let Ab b e the category of all ab elian gro ups and their homomorphisms. F ollowing [1 0 ], a fun ctori al top olo gy is a class τ = { τ A : A ∈ Ab } , wher e ( A, τ A ) is a top ological gro up for every A ∈ Ab , a nd every homomorphism in Ab is contin uous. In o t her words, a functoria l top o logy is a functor τ : Ab → T opAb such that τ ( A ) = ( A, τ A ) for every A ∈ Ab , where τ A denotes the topo logy on A , and τ ( φ ) = φ for every morphism φ in Ab [3]. F or an abe lia n gr oup G the profinite to p ology γ G and the Bohr top ology P G are functorial top ologies, as well as the natu r al top olo gy ν G , which ha s { mG : m ∈ N + } as a base of the neig h bo rhoo ds of 0 . Mor eo ver, the pro finit e a nd the natural topo logy are linear to p ologies. These three functorial top o logies are related b y the following equality prov ed in [6]: γ G = inf { ν G , P G } . (4.1) The se c o nd mo dification that we co nsider is the line ar mo dific ation τ λ of τ , that is, the group to p ology on G which ha s all the τ -op en subgroups as a base of the neig hbo rhoo ds o f 0. Lemma 4.2. F or G an ab elian gr oup, ( P G ) λ = γ G . Pr o of. Since γ G ≤ P G , w e have ( γ G ) λ ≤ ( P G ) λ . Mor e over, γ G = ( γ G ) λ and ( P G ) λ ≤ γ G . Prop osition 4. 3. L et ( G, τ ) and ( H, σ ) b e ab elian top olo gic al gr oups and φ : ( G, τ ) → ( H , σ ) a c ontinuous surje ct i ve homomorphi sm. (a) If τ is line ar and total ly b ounde d, then σ is line ar and total ly b ounde d as wel l. (b) If ( G, τ ) is b oun d e d torsion and total ly b ounde d, t h en τ is line ar. Pr o of. (a) It is clear that σ is totally b ounded. Assume first that τ is precompact. Since τ is linear and preco mpact, ^ ( G, τ ) is compact a nd linear. The extension e φ : ^ ( G, τ ) → ^ ( H, σ ) is contin uous and surjective, so op en as ^ ( G, τ ) is compact. Conseq uen tly , ^ ( H, σ ) is compact and linear. Now w e can conclude tha t σ is linea r a s well. If now τ is o nly totally bo und ed, consider the q uotien t ( G/ { 0 } τ , τ q ) of ( G, τ ). Then τ q is preco mpact, and it is line a r since τ is linear. The homomorphism φ : ( G/ { 0 } τ , τ q ) → ( G/ { 0 } σ , σ q ) induced b y φ is surjective and con tinuous. By the previo us case of the pro of we have that σ q is linear. Since σ is the initial topolo gy of σ q by the cano nical pro jection, w e can conclude that σ is linear as well. (b) Assume first that τ is precompact. Since ( G, τ ) is bounded torsion, its (co mpact) completion ^ ( G, τ ) is bo unded to rsion as well. Therefor e, ^ ( G, τ ) is compact and totally disconnected, so linear. If now τ is only bounded torsio n, consider the quotien t ( G/ { 0 } τ , τ q ) of ( G, τ ). This quotient is precompact and s o τ q is linear by the previous part of the pro of. Since τ is the initial top ology of τ q by the canonica l pro jection, τ is linea r a s well. W e collect in Lemma 4.4 some basic pr o perties of the combin atio n of the linear a nd the Bohr mo difications. Lemma 4.4. L et ( G, τ ) b e a top olo gic al ab elian gr oup. The n: (a) ( τ λ ) + = ( τ + ) λ , s o we c an write simply τ + λ ; (b) τ + λ is line ar and total ly b ounde d; (c) τ is line ar and total ly b ounde d if and only if τ = τ + λ ; (d) C τ + λ ( G ) = C τ λ ( G ) = C τ + ( G ) = C τ ( G ) ⊆ C ( G ) ; (e) C τ ( G ) is a b ase of the heighb ourho o ds of 0 in ( G, τ + λ ) ; 8 (f ) τ + λ = inf { τ , γ G } . Pr o of. (a), (b), (c), (d) and (e) are clear . (f ) Clearly , τ + λ ≤ inf { τ , γ G } . On the other ha nd, Pro p osition 4.3 ( a) implies that inf { τ , γ G } is linea r and totally b o unded, hence it coincides with τ + λ . In particular , we have the following diagram in the lattice of all gro up top ologies of an ab elian g roup G : τ | | | | | | | | C C C C C C C C γ G u u u u u u u u u u u u u u u u u u u u u u u u τ λ @ @ @ @ @ @ @ @ τ + } } } } } } } } τ + λ F or an ab elian g roup G , recall that G 1 = T n ∈ N + nG is the first U lm su b gr oup , whic h is fully inv ar ian t in G . It is well known that G 1 = T N ∈C ( G ) N [10]. F or a top ological ab elian group ( G, τ ), in analogy to the first Ulm subgr oup, define G 1 τ = \ N ∈ C τ ( G ) N , and let φ 1 τ : ( G/G 1 τ , τ q ) → ( G/G 1 τ , τ q ) be the contin uous endomorphis m induced by φ , where τ q is the quotient top ology of τ . Note that, unlik e the Ulm subg roup, its top ological version G 1 τ may fail to coincide with T { nG : n ∈ N + , nG ∈ C τ ( G ) } as the following ex a mple shows. Example 4. 5. Let G = Z ( p ) ( N ) for a prime p , a nd let τ b e the pro du ct to pology on G . Then G 1 τ = 0, while T { nG : n ∈ N + , nG ∈ C τ ( G ) } = G . The equality proved in Prop osition 4.7(a) b elo w shows the corr ect counterpart of the equality in the top ological c a se. Since G is residually finite if a nd only if G 1 = 0, w e say that G is τ -r esidual ly finite if G 1 τ = 0. Note that ( G/G 1 τ ) 1 τ q = 0. Clearly , for G an ab elian gr oup, γ G -residually finite means residually finite. F urthermore, G 1 τ ⊇ G 1 and so τ -residua lly finite implies r esidually finite. The following exa mple shows that the conv erse implication do es not hold true in general. Example 4.6. Let α ∈ T be an element of infinite order, co nsider the inclusion Z → T given by 1 7→ α and endow Z with the topo logy τ α inherited from T by this inclusio n. Then Z 1 = 0, while Z 1 τ α = Z . In particula r, Z is residually finite but not τ α -residually finite. Item (a) of Prop osition 4.7 gives a different description of G 1 τ , while item (b) shows the relation b et ween the dual of a top ological abelia n gro up ( G, τ ) a nd the dual o f ( G, τ + λ ); b o th pa rts use t (( G, τ ) ′ ). Prop osition 4.7. L et ( G, τ ) b e a top olo gic al ab elian gr oup. (a) Then G 1 τ = T χ ∈ t (( G,τ ) ′ ) ker χ . (b) If G is τ -r esidual ly finit e, then ( G, τ + λ ) ′ = t (( G, τ ) ′ ) . Pr o of. Let N ∈ C τ ( G ) and let χ ∈ ( G, τ ) ′ be such that χ ( N ) = 0. Then χ ∈ t (( G, τ ) ′ ). In fact, since G/ N is finite, there exists m ∈ N + such that mG ⊆ N . Then χ ( mG ) = 0 a nd so mχ = 0; in particular, χ ∈ t (( G, τ ) ′ ). (a) Let N ∈ C τ ( G ). Since G/ N is finite and discrete, G/ N is isomorphic to Z ( k 1 ) × . . . × Z ( k n ), for some k 1 , . . . , k n ∈ N + . F or each i = 1 , . . . , n , let χ i : Z ( k i ) → T be the embedding. Then χ = χ 1 × . . . × χ n gives an embedding G/ N → T n . Let now χ = χ ◦ π ∈ ( G, τ ) ′ , wher e π : G → G/ N is the cano nica l pro jection. So N = ker χ . By the starting observ ation in the pro of, χ ∈ t (( G, τ ) ′ ). This proves that G 1 τ ⊇ T χ ∈ t (( G,τ ) ′ ) ker χ . T o verify the conv erse inclusion it suffices to note that, if χ ∈ t (( G, τ ) ′ ), then ker χ ∈ C τ ( G ). (b) Fix a neighbor hoo d U of 0 in T that cont ains no non-zero subgr oups o f T . F or ev ery contin uous character χ : ( G, τ + λ ) → T , there exists N ∈ C τ ( G ) suc h that χ ( N ) ⊆ U . B y the c hoice of U this yields 9 χ ( N ) = 0, so χ ∈ t (( G, τ ) ′ by the starting o bserv ation in the pr oof. Hence, we hav e verified the inclusion ( G, τ + λ ) ′ ⊆ t (( G , τ ) ′ ). T o prov e the conv er s e inclusion, let χ ∈ t (( G, τ ) ′ ). W e hav e to verify that χ : ( G, τ + λ ) → T is contin uous. First note that χ : ( G, τ + ) → T is con tinuous. Mo reo ver, by h yp othesis there exists m ∈ N + such that mχ = 0, and so χ factorizes thro ugh the canonical pro jection π : ( G, τ + ) → ( G/ mG τ + , τ + q ), where τ + q is the q uotien t top ology of τ + , a nd the contin uous character χ : ( G/mG τ + , τ + q ) → T , that is , χ = χ ◦ π . Now ( G/mG τ + , τ + q ) is precompact and bounded tor sion, so linea r by Pr oposition 4.3(b). This yields that χ : ( G, τ + λ ) → T is contin uous, hence t (( G, τ ) ′ ) ⊆ ( G, τ + λ ) ′ , and this concludes the pro of. A consequence of this prop osition is that, in case G is a re sidually finite ab elian gr oup, then ( G, γ G ) ′ = t (Hom( G, T )); this equality is co n ta ine d in [6, Lemma 3.2 ]. Since φ † = b φ ↾ t ( b G ) , Theorem 1.2 yields en t ⋆ ( φ ) = ent ( b φ ↾ t ( b G ) ) = ent( φ † ), where φ † : ( G, γ G ) † → ( G, γ G ) † . In pa rticular, ent ⋆ ( φ ) = ent( φ † ). The la tt er equality will b e generalized b y Theorem 5.5. The next lemma giv es a characteriza t ion of τ -residua lly finite ab elian gr oups in terms of the mo dification τ + λ of τ . Lemma 4.8. F or a top olo gic al ab elian gr oup ( G, τ ) t h e fol lowing c onditions ar e e qu ivalent: (a) G is τ - r esidual ly finite; (b) G is τ + λ -r esidual ly fi nite; (c) τ + λ is Hausdorff (so pr e c omp act). In p articular, G/G 1 τ is τ q -r esidual ly fi nite, wher e τ q is the quotient top olo gy induc e d by τ on G/G 1 τ . Pr o of. (a) ⇔ (b) Since C τ ( G ) = C τ + λ ( G ), w e have that G 1 τ = G 1 τ + λ . (a) ⇔ (c) It suffices to note tha t G 1 τ = { 0 } τ + λ . Remark 4.9 . Let ( G, τ ) b e a top ological ab elian gro up and let φ : ( G, τ ) → ( G, τ ) be a contin uous endo- morphism. (a) T o understand where the topo logy τ + λ comes from, consider the B o hr compactificatio n b ( G, τ ) of ( G, τ ) with the canonical contin uous endomorphism  G : ( G, τ ) → b ( G, τ ). W e hav e the following diagram: b ( G, τ ) π   ( G, τ )  G 7 7 n n n n n n n n n n n n s ' ' P P P P P P P P P P P P id G   b ( G, τ ) /c ( b ( G, τ )) ( G, τ + λ ) s λ 7 7 n n n n n n n n n n n where π : b ( G, τ ) → b ( G, τ ) /c ( b ( G, τ )) is the cano nical pro jectio n, s = π ◦  G and s λ is s considere d on ( G, τ + λ ), that is, s = s λ ◦ id G . Then τ + λ is the initial top ology of the topolo gy of b ( G, τ ) /c ( b ( G, τ )), which is the (co mpact and to tally disco nnected) quotient topo logy of that of b ( G, τ ). (b) The von Neumann kernel ker  G is con tained in G 1 τ . Indeed, fo r N ∈ C τ ( G ), the quotien t G/ N is finite and discrete. So the character s s e parate the p oin ts of G/ N , and therefor e ker  G ⊆ N . Hence, ker  G ⊆ G 1 τ . (c) Assume that G is τ -r esidually finite. W e show that s (and so s λ ) is injective. Then  G is injectiv e b y item (b). Moreov er, s is injectiv e as well. In fact, ker  G = 0, and so we have k er s = G ∩ c ( b ( G, τ )). Since b ( G, τ ) is compact, c ( b ( G, τ )) = G 1 τ b . Ther efore, deno ted by τ b the top o logy of b ( G, τ ), ker s = G ∩ T N ∈C τ b ( b ( G,τ )) N = T N ∈C τ b ( b ( G,τ )) ( G ∩ N ) = T M ∈C τ ( G ) M = G 1 τ , and G 1 τ = 0 b y hypothesis . Hence, ker s = 0. 10 (d) If ( G, τ ) is preco mp ac t a nd G is τ -residually finite, then b ( G, τ ) = ^ ( G, τ ) and b ( G , τ ) /c ( b ( G, τ )) = ^ ( G, τ λ ). So the diagram in item (a) b ecomes: ^ ( G, τ ) π   ( G, τ ) (  6 6 l l l l l l l l l l l l l l l l id G   ^ ( G, τ λ ) = ^ ( G, τ ) /c ( ^ ( G, τ ) ) ( G, τ + λ ) )  6 6 m m m m m m m m m m m m m (4.2) where π = e id G . Note that the right hand side of the diagra m consis t s of the completions of the g roups on the left hand side. Now we pass to the ma in part o f this section, in which we give reductions for the computation of the top ological a djoin t en tropy . The next pro position gives a first top ological reduction for the co mput atio n of the top ological adjoint ent ro p y . Indeed, it shows that it is sufficient to cons ider gro up top ologies whic h are totally b ounded and linear. Note that if φ : ( G, τ ) → ( G, τ ) is a c o n tin uous endomorphism of a to pologica l abe lia n gro up ( G, τ ), then φ : ( G, τ + λ ) → ( G, τ + λ ) is contin uous as well, since, as noted ab o ve, b oth the B ohr modifica tion and the linear mo dification are functors T opAb → T opAb . Prop osition 4.1 0 . L et ( G, τ ) b e a top olo gic al ab elian gr oup and φ : ( G, τ ) → ( G, τ ) a c ontinuous endomor- phism. Then ent ⋆ τ ( φ ) = en t ⋆ τ λ ( φ ) = en t ⋆ τ + ( φ ) = en t ⋆ τ + λ ( φ ) . Pr o of. Since C τ ( G ) = C τ + λ ( G ) b y Lemma 4.4(d), b y the definition of topolo gical adjoint entropy it is pos s ible to conclude that ent ⋆ τ ( φ ) = en t ⋆ τ + λ ( φ ). T o prov e the other t wo equalities, it suffices to apply Lemma 3 .2 . The following result, whic h genera lizes [7 , Prop osition 4.1 3], is a nother reduction for the computatio n of the top ological adjoint en tropy . In fact, it shows that it is p ossible to restr ic t to τ -re s idually finite ab elian groups. Prop osition 4.1 1 . L et ( G, τ ) b e a top olo gic al ab elian gr oup and φ : ( G, τ ) → ( G, τ ) a c ontinuous endomor- phism. Then ent ⋆ τ ( φ ) = en t ⋆ τ q ( φ 1 τ ) , wher e τ q is the quotient top olo gy induc e d by τ on G/G 1 τ . Pr o of. By Lemma 3.8, ent ⋆ τ ( φ ) ≥ ent ⋆ τ q ( φ 1 τ ). Let N ∈ C τ ( G ). Then G 1 τ ⊆ N , and so N/ G 1 τ ∈ C τ q ( G/G 1 τ ). By (3.1), we k now that H ⋆ ( φ, N ) = H ⋆ ( φ 1 τ , N /G 1 τ ) ≤ ent ⋆ τ q ( φ 1 τ ). Then en t ⋆ τ ( φ ) ≤ en t ⋆ τ q ( φ 1 τ ), and hence ent ⋆ τ ( φ ) = en t ⋆ τ q ( φ 1 τ ). Remark 4.12. L e t ( G, τ ) b e a top ological ab elian gro up and let φ : ( G, τ ) → ( G, τ ) b e a contin uo us endomorphism. F o r the computation of the top ological adjoint en tropy of φ we ca n assume witho ut loss of generality that τ is precompact and linear. Indeed, b y Prop osition 4.11 we can assume that G is τ -re s idually finite. Mor eo ver, b y Lemma 4.8 this is equiv alent to say that τ + λ is precompact. Now, we can as sume without loss of generality that τ = τ + λ in view of Prop osition 4 .10, in other words, τ is pr ecompact and linea r. If in P ropositio n 4 .1 1 one co nsiders a φ -inv aria n t subgro up of G contained in G 1 τ instead of G 1 τ itself, then the sa me prop erty holds. In particular, every connected subg roup H of G is contained in G 1 τ , as each N ∈ C τ ( G ) is clop en and so N ⊇ H . So w e have the following consequence o f Prop osition 4.11, s ho wing the additivity of the topolo gical adjoint en tropy when o ne considers connected in v a rian t subgroups. 11 Corollary 4. 13. L et ( G, τ ) b e a top olo gic al ab elian gr oup, φ : ( G, τ ) → ( G, τ ) a c ontinuous endomorphism and H a φ -invariant su b gr oup of ( G, τ ) . If H is c onne cte d, t he n ent ⋆ τ ↾ H ( φ ↾ H ) = 0 and ent ⋆ τ ( φ ) = ent ⋆ τ q ( φ ) , wher e φ : ( G /H , τ q ) → ( G/H , τ q ) is the c ontinuous endomorphi sm induc e d by φ . In pa rticular, Co rollary 4 .13 ho lds fo r the co nn ected comp onen t o f ( G, τ ), namely , ent ⋆ τ ( φ ) = en t ⋆ τ q ( φ ), where φ : (( G, τ ) /c ( G, τ ) , τ q ) → (( G , τ ) /c ( G, τ ) , τ q ) is the cont inuous endomorphism induced b y φ . Therefore, the co mpu tation of the top ological a dj oint entropy ma y be reduced to the case o f contin uo us endomorphisms of totally disconnected ab elian groups. 5 T op ological adjoin t en trop y , top ological en trop y and algebraic en trop y Since our aim is to compar e the top ological a djo int en tropy with the topo logical entropy , we first recall the definition of top ological en tro p y given b y Adler, Konheim and McAndrew [1 ]. F or a compa ct top ological space X and for a n ope n cov er U of X , le t N ( U ) b e the minimal car dina lit y of a sub cov e r of U . Since X is compact, N ( U ) is alwa ys finite. Let H ( U ) = log N ( U ) be the entr opy of U . F or any tw o op e n cov ers U and V of X , let U ∨ V = { U ∩ V : U ∈ U , V ∈ V } . Define ana logously U 1 ∨ . . . ∨ U n , for op en cov e rs U 1 , . . . , U n of X . Let ψ : X → X b e a con tinuous map and U an op en cov er of X . Then ψ − 1 ( U ) = { ψ − 1 ( U ) : U ∈ U } . The top olo gic al entr opy of ψ with r esp e ct t o U is H top ( ψ , U ) = lim n →∞ H ( U ∨ ψ − 1 ( U ) ∨ . . . ∨ ψ − n +1 ( U )) n , and the top olo gic al entr opy o f ψ is h top ( ψ ) = sup { H top ( ψ , U ) : U o pen cov er of X } . Remark 5.1. P eters [16] modified the ab ov e definition o f alg ebraic entropy for automorphisms φ of arbitrary ab elian gro ups G using finite subse ts ins tead o f finite subg roups. In [5 ] this definition is extended in appropria te wa y to a ll endomo rphisms of ab elian gro ups, as follows. F o r a non-e mpty finite subset F of G and for a n y po sitiv e integer n , the n -th φ -tra je ctory of F is T n ( φ, F ) = F + φ ( F ) + . . . + φ n − 1 ( F ) . The limit H ( φ, F ) = lim n →∞ log | T n ( φ,F ) | n exists and is the algebr aic entr opy of φ with r esp e ct t o F . The algebr aic entr opy of φ is h ( φ ) = sup { H ( φ, F ) : F ⊆ G non-empty , finite } . In particular , ent ( φ ) = h ( φ ↾ t ( G ) ). In [5] we strengthen Theorem 1 .3, proving the following Bridge Theorem for the alg ebraic en tro p y h in the general setting of endomor phisms of ab elian groups: L et G b e an ab elian gr oup and φ ∈ End( G ) . The n h ( φ ) = h top ( b φ ) . W e see now that it is in the pr oof of W eiss ’ Bridge Theor em 1 .3 that one c a n find a first (non- explicit) app earance of the top ological adjoint entropy . Indeed, that pro of c on tains the following Lemma 5.2, which gives a “lo cal equality” betw een the top ological adjoint en tro py and the topo logical en tropy . In this sense it bec omes na tur al to introduce the top ological adjoint en tropy (and the adjoint algebr aic entrop y as in [7]). F ollowing [19, Section 2], for a top ological ab elian gr oup ( K, τ ) and C ∈ C τ ( G ), let ζ ( C ) = { x + C : x ∈ G } , which is an op en cov er of K . If K is precompact, then ζ ( C ) is finite, since ev ery ope n subgroup of ( K , τ ) has finite index in K . Lemma 5.2 . L et ( K, τ ) b e a c omp act ab elian gr oup and let ψ : ( K, τ ) → ( K , τ ) b e a c ontinuous endomo rphism. If C ∈ C τ ( K ) , then H ⋆ ( ψ , C ) = H top ( ψ , ζ ( C )) . 12 Pr o of. Let C ∈ C τ ( K ). Ob viously , N ( ζ ( C )) = [ K : C ]. Moreover, one can prov e by induction tha t ζ ( ψ − n ( C )) = ψ − n ( ζ ( C )) for every n ∈ N + and that ζ ( C 1 ) ∨ . . . ∨ ζ ( C n ) = ζ ( C 1 ∩ . . . ∩ C n ) for every n ∈ N + and C 1 , . . . , C n ∈ C τ ( K ). This implies that, fo r e very n ∈ N + , ζ ( C ) ∨ ψ − 1 ( ζ ( C )) ∨ . . . ∨ ψ − n +1 ( ζ ( C )) = ζ ( C ∩ ψ − 1 ( C ) ∩ . . . ∩ ψ − n +1 ( C )) = ζ ( B n ( ψ , C )) , so that N ( ζ ( C ) ∨ ψ − 1 ζ ( C ) ∨ . . . ∨ ψ − n +1 ζ ( C )) = N ( ζ ( C ∩ ψ − 1 ( C ) ∩ . . . ∩ ψ − n +1 ( C ))) = N ( ζ ( B n ( ψ , C ))) = [ K : B n ( ψ , C )] = | C n ( ψ , C ) | . Hence w e have the thesis. In view of this lemma we can connect the topo logical a djoin t ent ro p y and the top ological en tropy for contin uous endomorphisms of compact ab elian groups: Theorem 5.3. L et ( K , τ ) b e a c omp act ab elian gr oup and let ψ : ( K , τ ) → ( K, τ ) b e a c ontinuous endomor- phism. Then ent ⋆ τ ( ψ ) = ent ⋆ τ q ( ψ ) = h top ( ψ ) , wher e ψ : ( K/c ( K ) , τ q ) → ( K /c ( K ) , τ q ) is the c ontinuous endomorphism induc e d by ψ and τ q is the quotient top olo gy induc e d by τ on K/c ( K ) . In p articular, if ( K, τ ) is total ly disc onne cte d, then e n t ⋆ τ ( ψ ) = h top ( ψ ) . Pr o of. Corollary 4.13 gives en t ⋆ τ ( ψ ) = en t ⋆ τ q ( ψ ). Since ( K/ c ( K ) , τ q ) is to t ally disco nnected, ev ery op en cov er of K/c ( K ) is refined by a n open cov er of the form ζ ( C ), where C ∈ C τ q ( K/c ( K )). Hence, h top ( ψ ) = sup { H top ( ψ , ζ ( C )) : C ∈ C τ q ( K/c ( K )) } . (5.1) By Lemma 5.2, ent ⋆ τ q ( ψ ) = h top ( ψ ). The equality in (5.1) is contained also in the pr oof of [19, Theorem 2.1]. As a co rollary of Theorem 5 .3 and o f W eis s’ Bridge Theorem 1 .3 we obtain the following Bridge Theor em betw een the algebraic entropy and the top ological adjoint entropy in the general case of endo mo rphisms of ab elian g roups; this is Theorem 1.4 of the int ro duction. Corollary 5.4 (Bridge Theor e m ) . L et ( K, τ ) b e a c omp act ab elian gr oup and ψ : ( K, τ ) → ( K, τ ) a c ontinuous endomorphi sm. Then ent ⋆ τ ( ψ ) = ent( b ψ ) . Pr o of. Let G = b K and φ = b ψ . By the definition o f a lgebraic entrop y a nd by Theorem 5.3 w e ha ve r e spectively ent ( φ ) = en t( φ ↾ t ( G ) ) and ent ⋆ b τ ( ψ ) = ent ⋆ b τ q ( ψ ) , (5.2) where ψ : ( K /c ( K ) , b τ q ) → ( K /c ( K ) , b τ q ) is the con tinuous endomo rphism induced by ψ and b τ q is the quotien t top ology of b τ . By the Pon tryagin duality d t ( G ) is top ologically isomorphic to K /c ( K ) and \ φ ↾ t ( G ) is c o njugated to ψ by a topo logical isomo rphism. The topo logical en tro p y is inv aria nt under conjugation by top ological isomorphisms, so h top ( \ φ ↾ t ( G ) ) = h top ( ψ ) . (5.3) Now Theor em 1.3 and Theor em 5.3 give respectively ent ( φ ↾ t ( G ) ) = h top ( \ φ ↾ t ( G ) ) and ent ⋆ b τ q ( ψ ) = h top ( ψ ) . (5.4) The thesis follows from (5.2), (5.3) and (5.4). 13 This Bridg e Theor em shows that the topolog ical a djo in t entropy ent ⋆ τ is the top ological counterpart of the algebraic entrop y ent, as well as the top ological entrop y h top is the top ological counterpart of the alg ebraic ent ro p y h in view of the Br idge Theorem stated in Rema rk 5.1. Indeed, r oughly sp eaking, ent ⋆ τ reduces to totally disco nn ected compact ab elian groups, a s dually the algebra ic ent ro p y en t reduces to tor sion abe lian groups. The following is another Bridge Theore m, s ho wing that the to pologica l adjo int en tro p y of a contin uo us endomorphism of a preco mpact abelian group is the same as the algebr aic entropy of the dual endomorphism in the precompact duality . Its Co rollary 5.6 will show that this e ntropy coincides also with the alg ebraic ent ro p y o f the dual endomorphism in the Pon tryagin duality of its extension to the completion. Theorem 5 .5 (Bridge Theorem) . L et ( G, τ ) b e a pr e c omp act ab elian gr oup and φ : ( G, τ ) → ( G, τ ) a c ontin- uous endomorph ism. Then ent ⋆ τ ( φ ) = ent( φ † ) . Pr o of. Let N ∈ C τ ( G ). Then F = N ⊥ is a finite subg roup of ( G, τ ) † by F act 2 .1 (a). By F ac t 2.1(c) ( φ − n ( N )) ⊥ = ( φ † ) n ( F ) for every n ∈ N . Hence, B n ( φ, N ) ⊥ = T n ( φ † , F ) for every n ∈ N + by F act 2.1(b). It follows that | C n ( φ, N ) | = | C n ( φ, N ) † | = | B n ( φ, N ) ⊥ | = | T n ( φ † , F ) | for every n ∈ N + , and this concludes the pro of. Corollary 5.6. L et ( G, τ ) b e a pr e c omp act ab elian gr oup. Denote by e τ t h e top olo gy of the c ompletion K = ^ ( G, τ ) , let e φ : K → K b e the ex t ensio n of φ , and let e φ : K/c ( K ) → K/c ( K ) b e the endomorphi sm induc e d by e φ . The n ent ( φ † ) = ent ⋆ τ ( φ ) = en t ⋆ e τ ( e φ ) = en t( b e φ ) = h top ( e φ ) . Pr o of. The first equality is The o rem 5.5, the se cond is Co rollary 3.12, while the equa lit y ent ⋆ e τ ( e φ ) = h top ( e φ ) follows from Theo rem 5.3 and ent( b e φ ) = h top ( e φ ) from Theore m 1.3. In the h yp otheses o f Coro llary 5.6, assume that ( G, τ ) is linea r. Then ^ ( G, τ ) is linear a s well and we obtain ent ( φ † ) = ent ⋆ τ ( φ ) = en t ⋆ e τ ( e φ ) = en t( b e φ ) = h top ( e φ ) . If one as s umes only that ( G, τ ) is a totally disconnected compac t a belian gro up, then ^ ( G, τ ) is no t tota lly disconnected in general (it can be connected), and the equality ent ⋆ e τ ( e φ ) = h top ( e φ ) may fail a s the following example shows. Example 5.7. Let G = Z and denote by e τ the compact topo logy of T . Let α ∈ T be a n ele ment of infinite order, c onsider the inclusion Z → T given by 1 7→ α and endow Z with the to pology τ α inherited fro m T by this inclusion. Then ( G, τ α ) is dense in T . Moreov er, consider e φ = µ 2 : T → T , defined by e φ ( x ) = 2 x for ev ery x ∈ T , a nd φ = µ 2 ↾ G . Now ent ⋆ τ α ( φ ) = 0, by Prop osition 5.5 (b) we ha ve en t ⋆ τ ( e φ ) = en t ⋆ τ α ( φ ), and so ent ⋆ e τ ( e φ ) = 0. On the other hand, the Kolmog oro v-Sina i Theo rem for the topo logical en tropy of automo rphisms of T n (see [13, 18]) gives h top ( e φ ) = log 2. The situation descr ibed by Example 5.7 can b e generalized as follows. Suppo se that ( G, τ ) is a precompact ab elian group s uc h that G is τ -r esidually finite. Let φ : ( G, τ ) → ( G, τ ) b e a contin uous endomorphism a nd e φ : ^ ( G, τ ) → ^ ( G, τ ) the extension o f φ to ^ ( G, τ ) . Since G is τ -res idually finite, τ λ = τ + λ is precompact by Lemma 4.8. Denote b y φ λ the endomor phism φ considered on ( G, τ λ ). Then φ λ : ( G, τ λ ) → ( G, τ λ ) is contin uous, and let e φ λ : ^ ( G, τ λ ) → ^ ( G, τ λ ) b e the extension of φ to ^ ( G, τ λ ). Denote b y e τ and e τ λ resp ectiv ely 14 the top ology o f ^ ( G, τ ) and ^ ( G, τ λ ). The situatio n is descr ibed by the following diagra m. ^ ( G, τ ) e φ / /   ^ ( G, τ ) e id G = π   ( G, τ ) φ / / ,  : : v v v v v v v v v id G   ( G, τ ) (  6 6 l l l l l l l l l l l l l l l   ^ ( G, τ λ ) e φ λ / / ^ ( G, τ λ ) = ^ ( G, τ ) /c ( ^ ( G, τ ) ) ( G, τ λ ) φ λ / / ,  : : v v v v v v v v v ( G, τ λ ) (  6 6 l l l l l l l l l l l l l The right side face of the cubic diag ram (which obviously co incides with the le f t o ne) is describ ed by (4.2), that is, ^ ( G, τ λ ) = ^ ( G, τ ) /c ( ^ ( G, τ ) ) a nd the extension e id G of id G : ( G, τ ) → ( G, τ λ ) coincides with the canonical pro jection π : ^ ( G, τ ) → ^ ( G, τ ) /c ( ^ ( G, τ ) ). So in the cubic diagra m we ha ve four cont inuous endomorphisms, namely φ , φ λ , e φ and e φ λ , that hav e the same top ological a dj oint entrop y; indeed, P r oposition 4.10 cov ers the fr o n t face of the cub e (i.e., ent ⋆ τ ( φ ) = ent ⋆ τ λ ( φ λ )), while Cor ollary 3.1 2 applies to the upp er and the low er faces (i.e., ent ⋆ τ ( φ ) = ent ⋆ e τ ( e φ ) and ent ⋆ τ λ ( φ λ ) = ent ⋆ e τ λ ( e φ λ )). On the other hand w e can consider the top o logical entrop y only for e φ and e φ λ , as the completions are compact. By Theor e m 5.3, h top ( e φ λ ) = ent e τ λ ( φ λ ) and so h top ( e φ λ ) coincides with the top ological adjoin t ent ro p y of the four contin uous endomorphisms. F or the to p ological entropy of e φ we can s a y only that h top ( e φ ) ≥ h top ( e φ λ ), in view of the monoto nicit y pro perty of the top ological entrop y . W e co llect a ll these obser v a tio ns in the following formula: h top ( e φ ) ≥ h top ( e φ λ ) = ent ⋆ e τ λ ( e φ λ ) = ent ⋆ τ λ ( φ λ ) = ent ⋆ τ ( φ ) = ent ⋆ e τ ( e φ ) . The nex t example shows that this inequa lit y can b e str ict in the strongest w ay; indeed, we find a φ such that h top ( e φ ) = ∞ and h top ( e φ λ ) = 0 . Example 5 . 8. Let G = L p ∈ P Z ( p ). Since G is residually finite, γ G is Hausdorff b y Lemma 4.8, and s o P G is Hausdorff as well, as γ G ≤ P G (see (4.1)). Moreov er, γ G = ν G . Let µ 2 denote the endomorphism given b y the multiplication b y 2, and set φ = µ 2 : G → G . Consider φ : ( G, P G ) → ( G, P G ). Then φ can b e extended to the Bo hr compa ctification K = bG of G ( K coincides with the completion of ( G, P G )) and let e φ : K → K b e this extension. By the densit y of G in K , and the uniqueness of the extensio n, e φ = µ 2 : K → K . By [2, Theorem 3.5 - Case 1 of the pro of] w ith α = µ 2 , we hav e h top ( e φ ) = ∞ . Consider now φ λ : ( G, γ G ) → ( G, γ G ), where ( P G ) λ = γ G by Lemma 4.2. The completion K λ of ( G, γ G ) is K λ = Q p ∈ P Z ( p ), a s γ G = ν G , and φ extends to e φ λ : K λ → K λ ; de no te by e γ G the topolo gy of K λ . Moreov er , ent ⋆ e γ G ( e φ λ ) = 0 , b ecause every finite-index s ubgroup o f K λ is e φ λ -inv a rian t. By Theorem 5.3 w e ha ve h top ( e φ λ ) = ent ⋆ e γ G ( e φ λ ) and so h top ( e φ λ ) = 0 . 6 The top ological adjoin t en trop y of the Bernoulli shifts In the following ex a mple w e r ecall the pro of given in [7] of the fact that the adjoint a lgebraic en tropy of the Bernoulli shifts is infinite. Example 6. 1. If K is a non-trivia l ab elian group, then ent ⋆ ( β ⊕ K ) = ent ⋆ ( K β ⊕ ) = ent ⋆ ( β ⊕ K ) = ∞ . 15 Indeed, by [11, Corollar y 6.5], ent( β K ) = en t( K β ) = ent( β K ) = ∞ . Moreov er, c β ⊕ K = K β , [ K β ⊕ = β K , and c β ⊕ K = ( β K ) − 1 by [7, Pr oposition 6.1]. Then Theo r em 1 .2 implies ent ⋆ ( β ⊕ K ) = ent( K β ) = ∞ , ent ⋆ ( K β ⊕ ) = ent ( β K ) = ∞ and en t ⋆ ( β ⊕ K ) = ent(( β K ) − 1 ) = ∞ . W e give no w a direct computation o f the v alue o f the algebra ic adjoint entropy of the Bernoulli shifts. The starting idea for this pro of w as given to me b y Brenda n Goldsmith and Keta o Gong. Prop osition 6.2. L et p b e a prime and K = Z ( p ) . Then ent ⋆ ( β ⊕ K ) = ent ⋆ ( K β ⊕ ) = ent ⋆ ( β ⊕ K ) = ∞ . Pr o of. Since K ( N ) ∼ = K ( N + ) , we consider without lo ss o f genera lit y K ( N + ) instead of K ( N ) , since it is co n venient for our pro of. W rite K ( N ) = L n ∈ N + h e n i , where { e n : n ∈ N + } is the canonica l base of K ( N ) . (i) W e start proving that ent ⋆ ( β ⊕ K ) = ∞ . Define, for i ∈ N , δ 1 ( i ) = ( 1 if i = (2 n )! + n for so me n ∈ N + , 0 otherwise . δ 2 ( i ) = ( 1 if i = (2 n + 1)! + n for some n ∈ N + , 0 otherwise . Let N 2 = h e i − δ 1 ( i ) e 1 − δ 2 ( i ) e 2 : i ≥ 3 i ∈ C ( K ( N ) ). Then e 1 , e 2 6∈ N 2 and K ( N ) = N 2 ⊕ h e 1 i ⊕ h e 2 i . F o r every n ∈ N + , h e (2 n )! , e (2 n +1)! i ⊆ B n ( β ⊕ K , N 2 ); (6.1) indeed, ( β ⊕ K ) k ( e (2 n )! ) = e (2 n )!+ k ∈ N 2 and ( β ⊕ K ) k ( e (2 n +1)! ) = e (2 n +1)!+ k ∈ N 2 for every k ∈ N with k < n . Moreov er , h e (2 n )! , e (2 n +1)! i ∩ B n +1 ( β ⊕ K , N 2 ) = 0 . (6.2) In fact, s ince h e (2 n )! , e (2 n +1)! i ⊆ B n ( β ⊕ K , N 2 ), w e hav e a 1 e (2 n !) + a 2 e (2 n +1)! ∈ B n +1 ( β ⊕ K , N 2 ) for a 1 , a 2 ∈ Z ( p ) if and only if ( β ⊕ K ) n ( a 1 e (2 n !) + a 2 e (2 n +1)! ) = a 1 e (2 n !+ n ) + a 2 e (2 n +1)!+ n ∈ N 2 ; since e (2 n !)+ n − e 1 ∈ N 2 and e (2 n +1)!+ n − e 2 ∈ N 2 , this is equiv a len t to a 1 e 1 + a 2 e 2 ∈ N 2 , which o ccurs if and only if a 1 = a 2 = 0. By (6 .1 ) and (6.2), for ev er y n ∈ N + ,     B n ( β ⊕ K , N 2 ) B n +1 ( β ⊕ K , N 2 )     ≥ p 2 . So (1.2) gives H ( β ⊕ K , N 2 ) ≥ 2 log p . Generalizing this ar gumen t, fo r m ∈ N + , m > 1 , define, fo r i ∈ N , δ 1 ( i ) = ( 1 if i = ( mn )! + n for some n ∈ N + , 0 otherwise . δ 2 ( i ) = ( 1 if i = ( mn + 1)! + n for so me n ∈ N + , 0 otherwise . . . . δ m ( i ) = ( 1 if i = ( mn + m − 1 ) ! + n for so me n ∈ N + , 0 otherwise . Let N m = h e i − δ 1 ( i ) e 1 − . . . − δ m ( i ) e m : i ≥ m + 1 i ∈ C ( K ( N ) ). Then e 1 , . . . , e m 6∈ N m and K ( N ) = N m ⊕ h e 1 i ⊕ . . . ⊕ h e m i . F o r every n ∈ N + , h e ( mn )! , . . . , e ( mn + m − 1)! i ⊆ B n ( β ⊕ K , N m ); (6.3) indeed, ( β ⊕ K ) k ( e ( mn )! ) = e ( mn )!+ k ∈ N m , . . . , ( β ⊕ K ) k ( e ( mn + m − 1)! ) = e ( mn +1)!+ m − 1 ∈ N m for ev ery k ∈ N with k < n . 16 Moreov er , h e ( mn )! , e ( mn +1)! i ∩ B n +1 ( β ⊕ K , N m ) = 0 . (6.4) In fact, since h e ( mn )! , . . . , e ( mn + m − 1)! i ⊆ B n ( β ⊕ K , N m ), we hav e a 1 e ( mn !) + . . . + a m e ( mn + m − 1)! ∈ B n +1 ( β ⊕ K , N m ) for a 1 , . . . , a m ∈ Z ( p ) if and only if ( β ⊕ K ) n ( a 1 e ( mn !) + . . . + a m e ( mn + m − 1)! ) = a 1 e ( mn !+ n ) + . . . + a m e ( mn + m − 1)!+ n ∈ N m ; since e ( mn !)+ n − e 1 ∈ N m , . . . , e ( mn + m − 1)!+ n − e m ∈ N m , this is equiv alent to a 1 e 1 + . . . + a m e m ∈ N m , which o ccurs if and only if a 1 = . . . = a m = 0. By (6 .3 ) and (6.4), for ev er y n ∈ N + ,     B n ( β ⊕ K , N m ) B n +1 ( β ⊕ K , N m )     ≥ p m . So (1.2) yields H ( β ⊕ K , N m ) ≥ m log p . W e have seen that ent ⋆ ( β ⊕ K ) ≥ m log p for every m ∈ N + and so ent ⋆ ( β ⊕ K ) = ∞ . (ii) The same arg um ent as in (i) shows that ent ⋆ ( β ⊕ K ) = ∞ . (iii) An analog o us ar gumen t shows that ent ⋆ ( K β ⊕ ) = ∞ . Indeed, it suffices to define, for every m ∈ N + , m > 1, δ ′ 1 ( i ) = ( 1 if i = ( mn )! − n for some n ∈ N + , 0 otherwise . δ ′ 2 ( i ) = ( 1 if i = ( mn + 1)! − n for so me n ∈ N + , 0 otherwise . . . . δ ′ m ( i ) = ( 1 if i = ( mn + m − 1 ) ! − n for so me n ∈ N + , 0 otherwise , N m = h e i − δ ′ 1 ( i ) e 1 − . . . − δ ′ m ( i ) e m : i ≥ m + 1 i ∈ C ( K ( N ) ), and pro ceed a s in (i). Example 6. 3. Let K b e a non-trivial abelian group. Example 6.1 (and s o Pr oposition 6.2) is equiv alent to en t ⋆ γ K ( N ) ( β ⊕ K ) = ent ⋆ γ K ( N ) ( K β ⊕ ) = ent ⋆ γ K ( Z ) ( β K ) = ∞ . On the other hand, we can consider the top ology τ K on K ( N ) (resp ectiv ely , K ( Z ) ) induced by the pro duct top ology o f K N (resp ectiv ely , K Z ). Cle a rly , τ K ≤ γ K ( N ) (resp ectiv ely , τ K ≤ γ K ( Z ) ). Moreover, C τ K ( K ( N ) ) = { 0 F ⊕ K ( N \ F ) : F ⊆ N , F finite } (resp ectiv ely , C τ K ( K ( Z ) ) = { 0 F ⊕ K ( Z \ F ) : F ⊆ Z , F finite } ). Then we see that (i) ent ⋆ τ K ( β ⊕ K ) = 0 a nd (ii) ent ⋆ τ K ( K β ⊕ ) = ent ⋆ τ K ( β ⊕ K ) = lo g | K | . Indeed, every N ∈ C τ K ( K ( N ) ) contains N m = 0 ⊕ . . . ⊕ 0 | {z } m ⊕ K N \{ 0 ,...,m − 1 } for some m ∈ N + . By Lemma 3.1 it s uffices to calculate the topo logical adjoint entrop y with resp ect to the N m . (i) Let m ∈ N + . F or every n ∈ N + , B n ( β ⊕ K , N m ) = N m and so H ( β ⊕ K , N m ) = 0 . Therefore, en t ⋆ τ K ( β ⊕ K ) = 0 . (ii) Let m ∈ N + . F or every n ∈ N + , B n ( K β ⊕ , N m ) = N m + n − 1 . So | C n ( K β ⊕ , N m ) | = | K | m + n − 1 and hence H ⋆ ( K β ⊕ , N m ) = log | K | . Consequently , ent ⋆ τ K ( K β ⊕ ) = log | K | . The pro of that ent ⋆ τ K ( β ⊕ K ) = log | K | is analogo us. 17 References [1] R. L. Adler, A. G. Konheim and M. H . McAndrew, T op olo gic al entr opy , T rans. Amer. Math. Soc. 114 (1965), 309–319 . [2] D. Alcaraz, D. Dikranjan and M. Sanchis, Infinitude of Bowen ’s entr opy for gr oup endomorphisms , prep ri nt. 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