Topological Dynamics indexed by words

TOPOLOGICAL D YNAMICS INDEXED BY W ORDS V ASSILIKI F ARMAKI AND ANDREAS KOUTSOGIANNIS Abstract. Starting with a combinatorial partition theorem for w ords ov er an infinite alphab et domina ted by a fixed sequence, es tablished recently b y the authors, we prov e recurrence results for top ological dynamical systems indexed b y suc h w ords. In this w ay we extend t he classical theory de velop ed by F urstenberg and W eiss of dynamical sys tems indexed by the natural num b ers to systems indexe d by w ords. Moreov er, applying this theory to top ological sy stems indexed by semigro ups that can be represented as words we get analogous recurrence results for such systems. Intr oduction F ursten b erg in collab oration with W e iss and Katznelson in the 1970’s ([F u], [F uW], [F uKa]) conne cted fundamen tal com binatorial results, s uc h as the partition theorems o f v an der W aerden ([vdW], 1927) and Hindman ([H], 1974), with top ological dynamics and particularly with phenomena of ( multiple) recurrence for suitable seque nces of contin uous functions defined on a compact metric space in to itself. The t heorems of v an der W aerden a nd Hindman w ere unified b y a partitio n theorem fo r w or ds o v er a finite alphab et of Carlson ([C], 1988); recen tly Carlson’s theorem w as essen- tially strengthened b y the authors, in [F], [FK], to a partition theorem (Theorem 1.2) for ω - Z ∗ -lo cated w or ds (i.e. words ov er an infinite alphab et dominated by a fixed sequence). Our starting p oint in this w ork is a top ological f o rm ulatio n of the partit ion theorem for ω - Z ∗ -lo cated w ords (Theorem 2.1). In tro ducing the notion of a dynamical sys tem o f con t in uous maps (homeomorphisms in the multiple case) fr o m a compact metric space in to itself indexed b y ω - Z ∗ -lo cated w ords, w e apply this form ulation to study (multiple) recurrence phenomena for these top olog ical systems (Theorems 2.6 , 2.15), extending the earlier results of Bir khoff ([Bi]) and F ur stenberg-W eiss ([F u], [F uW]). By making use of the represen tation o f rational and in teger num b ers as ω - Z ∗ -lo cated w or ds (Example 1.1) established b y Budak-I¸ sik-Pym in [BIP], w e obtain recurrence 1991 Mathematics Subje ct Classific ation. Pr ima ry 37Bxx; 54H20. Key wor ds and p hr ases. T opolo gical dyna mics , Ramsey theory , ω - Z ∗ -lo cated w ords, rational n umbers, IP-limits. 1 results for dynamical systems indexed b y ra tional n um b ers or by the in tegers (T heo- rems 3.1, 3.2, 3.3, 3.4 ). Moreo v er, we p oin t out the wa y to o btain recurrence results for dynamical sys tems indexed by an arbitr ary semigroup ( Theorems 3.5, 3.7). W e will use the follo wing nota t io n. Notation. L et N = { 1 , 2 , . . . } b e the set of natur al numb ers, Z = { . . . , − 2 , − 1 , 0 , 1 , 2 , . . . } the set of inte ger numb ers, Q = { m n : m ∈ Z , n ∈ N } the set of r a tional n umb ers and Z − = {− n : n ∈ N } , Z ∗ = Z \ { 0 } , Q ∗ = Q \ { 0 } . 1. A p ar t ition theorem for ω - Z ∗ -loca ted words In this section w e will introduce the ω - Z ∗ -lo cated words and w e will state a partition theorem for these words prov ed in [FK]. An ω - Z ∗ -lo cated w ord o v er the alphab et Σ = { α n : n ∈ Z ∗ } dominated b y ~ k = ( k n ) n ∈ Z ∗ , whe re k n ∈ N for ev ery n ∈ Z ∗ and ( k n ) n ∈ N , ( k − n ) n ∈ N are increasing sequences , is a function w from a non-empty , finite subset F of Z ∗ in to the alphab et Σ suc h that w ( n ) = w n ∈ { α 1 , . . . , α k n } for ev ery n ∈ F ∩ N and w n ∈ { α − k n , . . . , α − 1 } for ev ery n ∈ F ∩ Z − . So, the set e L (Σ , ~ k ) of all (constant) ω - Z ∗ -lo cated words o v er Σ do minated b y ~ k is: e L (Σ , ~ k ) = { w = w n 1 . . . w n l : l ∈ N , n 1 < . . . < n l ∈ Z ∗ and w n i ∈ { α 1 , . . . , α k n i } if n i > 0, w n i ∈ { α − k n i , . . . , α − 1 } if n i < 0 for ev ery 1 ≤ i ≤ l } . Analogously , the set of ω -lo cated w ords o v er the alphab et Σ = { α n : n ∈ N } dominated b y the increasing sequenc e ~ k = ( k n ) n ∈ N ⊆ N is L (Σ , ~ k ) = { w = w n 1 . . . w n l : l ∈ N , n 1 < . . . < n l ∈ N and w n i ∈ { α 1 , . . . , α k n i } for ev ery 1 ≤ i ≤ l } . Example 1.1. W e will giv e s ome ex amples of sets t ha t can b e represe n ted as ω - Z ∗ - lo cated w ords. (1) According to Budak-I¸ sik-Pym in [BIP], ev ery rational num b er q has a unique expression in the for m q = ∞ X s =1 q − s ( − 1) s ( s + 1)! + ∞ X r =1 q r ( − 1) r +1 r ! where ( q n ) n ∈ Z ∗ ⊆ N ∪ { 0 } with 0 ≤ q − s ≤ s fo r ev ery s > 0, 0 ≤ q r ≤ r for ev ery r > 0 and q − s = q r = 0 for all but finite many r , s . Setting Σ = { α n : n ∈ Z ∗ } , where α − n = α n = n for n ∈ N , and ~ k = ( k n ) n ∈ Z ∗ , where k − n = k n = n for n ∈ N , the function 2 g − 1 : Q ∗ → e L (Σ , ~ k ) , whic h sends q to the w or d w = q t 1 . . . q t l ∈ e L (Σ , ~ k ), where { t 1 , . . . , t l } = { t ∈ Z ∗ : q t 6 = 0 } , is one-to-one and onto. (2) According to [BIP ], for a given increasing sequenc e ( k n ) n ∈ N ⊆ N with k n ≥ 2, ev ery in teger n umber z ∈ Z has a unique expres sion in the form z = ∞ X s =1 z s ( − 1) s − 1 l s − 1 where l 0 = 1 , l s = k 1 . . . k s , fo r s ∈ N and ( z s ) s ∈ N ⊆ N ∪ { 0 } with 0 ≤ z s ≤ k s for ev ery s ∈ N and z s = 0 for all but finite man y s. Setting Σ = { α n : n ∈ N } , where α n = n , and ~ k = ( k n ) n ∈ N the function g − 1 : Z ∗ → L (Σ , ~ k ) , whic h s ends z to t he w ord w = z s 1 . . . z s t ∈ L (Σ , ~ k ), wh ere { s 1 , . . . , s t } = { s ∈ N : z s 6 = 0 } , is one-to-one and onto. (3) F or a giv en natural n um b er k > 1 , ev ery na tural n um b er n has a unique expression in the for m n = ∞ X s =1 n s k s − 1 where ( n s ) s ∈ N ⊆ N ∪ { 0 } with 0 ≤ n s ≤ k − 1 and n s = 0 for all but finite many s. Setting Σ = { 1 , . . . , k − 1 } and ~ k = ( k n ) n ∈ N with k n = k − 1 the function g − 1 : N → L (Σ , ~ k ) , whic h sends n to the w ord w = n s 1 . . . n s l ∈ L (Σ , ~ k ) , where { s 1 , . . . , s l } = { s ∈ N : n s 6 = 0 } , is one-to-one and onto. Let Σ = { α n : n ∈ Z ∗ } be an alphab et, ~ k = ( k n ) n ∈ Z ∗ ⊆ N suc h that ( k n ) n ∈ N , ( k − n ) n ∈ N are increasing seq uences and υ / ∈ Σ b e an en tity which is called a v ariable . The set of v ariable ω - Z ∗ -lo cated words o v er Σ dominated b y ~ k is: e L (Σ , ~ k ; υ ) = { w = w n 1 . . . w n l : l ∈ N , n 1 < . . . < n l ∈ Z ∗ , w n i ∈ { υ , α 1 , . . . , α k n i } if n i > 0, w n i ∈ { υ , α − k n i , . . . , α − 1 } if n i < 0 for all 1 ≤ i ≤ l and there exists 1 ≤ i ≤ l with w n i = υ } . The s et of v ariable ω -lo cated w ords o v er Σ = { α n : n ∈ N } dominated b y the increasing seq uence ~ k = ( k n ) n ∈ N ⊆ N is: L (Σ , ~ k ; υ ) = { w = w n 1 . . . w n l : l ∈ N , n 1 < . . . < n l ∈ N , w n i ∈ { υ , α 1 , . . . , α k n i } for all 1 ≤ i ≤ l and there exis ts 1 ≤ i ≤ l with w n i = υ } 3 W e set e L (Σ ∪ { υ } , ~ k ) = e L (Σ , ~ k ) ∪ e L (Σ , ~ k ; υ ) and L (Σ ∪ { υ } , ~ k ) = L (Σ , ~ k ) ∪ L (Σ , ~ k ; υ ). F or w = w n 1 . . . w n l ∈ e L (Σ ∪ { υ } , ~ k ) the set dom ( w ) = { n 1 , . . . , n l } is the domain of w . Let dom − ( w ) = { n ∈ do m ( w ) : n < 0 } and dom + ( w ) = { n ∈ do m ( w ) : n > 0 } . W e define the se t e L 0 (Σ , ~ k ; υ ) = { w ∈ e L (Σ , ~ k ; υ ) : w i 1 = υ = w i 2 for some i 1 ∈ dom − ( w ) , i 2 ∈ dom + ( w ) } . F or w = w n 1 . . . w n r , u = u m 1 . . . u m l ∈ e L (Σ ∪ { υ } , ~ k ) with dom ( w ) ∩ dom ( u ) = ∅ w e define the concatenating word : w ⋆ u = z q 1 . . . z q r + l ∈ e L (Σ ∪ { υ } , ~ k ) , where { q 1 < . . . < q r + l } = dom ( w ) ∪ d om ( u ) , z i = w i if i ∈ dom ( w ) and z i = u i if i ∈ dom ( u ). The set e L (Σ ∪ { υ } , ~ k ) can b e endo w ed with the relations < R 1 , < R 2 : w < R 1 u ⇐ ⇒ dom ( u ) = A 1 ∪ A 2 with A 1 , A 2 6 = ∅ suc h tha t max A 1 < min d om ( w ) ≤ max dom ( w ) < min A 2 , w < R 2 u ⇐ ⇒ ma x dom ( w ) < min dom ( u ) . W e define the se ts e L ∞ (Σ , ~ k ; υ ) = { ~ w = ( w n ) n ∈ N : w n ∈ e L 0 (Σ , ~ k ; υ ) and w n < R 1 w n +1 for ev ery n ∈ N } , L ∞ (Σ , ~ k ; υ ) = { ~ w = ( w n ) n ∈ N : w n ∈ L (Σ , ~ k ; υ ) and w n < R 2 w n +1 for ev ery n ∈ N } . W e will define now the notion of substitution for t he v aria ble ω - Z ∗ -lo cated w ords and respectiv ely f or the v aria ble ω -lo cated words. Let w = w n 1 . . . w n l ∈ e L 0 (Σ , ~ k ; υ ) with n w = min dom + ( w ) and − m w = max dom − ( w ) for n w , m w ∈ N . F o r eve ry ( p, q ) ∈ { 1 , . . . , k n w } × { 1 , . . . , k − m w } ∪ { ( υ , υ ) } we set: w ( υ , υ ) = w and w ( p, q ) = u n 1 . . . u n l , for ev ery ( p , q ) ∈ { 1 , . . . , k n w } × { 1 , . . . , k − m w } , where, for 1 ≤ i ≤ l , u n i = w n i if w n i ∈ Σ, u n i = α p if w n i = υ , n i > 0 and u n i = α − q if w n i = υ , n i < 0. Resp ectiv ely , Let w = w n 1 . . . w n l ∈ L (Σ , ~ k ; υ ) with n w = min dom ( w ) ∈ N . F or ev ery p ∈ { 1 , . . . , k n w } ∪ { υ } we se t: w ( υ ) = w and w ( p ) = u n 1 . . . u n l , for ev ery p ∈ { 1 , . . . , k n w } , where, for 1 ≤ i ≤ l , u n i = w n i if w n i ∈ Σ, u n i = α p if w n i = υ . W e r emark that for ~ w = ( w n ) n ∈ N ∈ e L ∞ (Σ , ~ k ; υ ) (resp. fo r ~ w = ( w n ) n ∈ N ∈ L ∞ (Σ , ~ k ; υ )) w e ha v e n ≤ min dom + ( w n ) and − n ≥ max dom − ( w n ) (resp. n ≤ min dom ( w n )), for n ∈ N . So, f o r n ∈ N , the substituted word w n ( p, q ) (resp. w n ( p ) ) has meaning f or ev ery ( p, q ) ∈ N × N with p ≤ k n and q ≤ k − n (resp. for ev ery p ∈ N with p ≤ k n ). 4 Fix a se quence ~ w = ( w n ) n ∈ N ∈ e L ∞ (Σ , ~ k ; υ ) (resp. ~ w = ( w n ) n ∈ N ∈ L ∞ (Σ , ~ k ; υ )). An extracted ω - Z ∗ -lo cated word (resp. extract ed ω -lo cated w ord ) of ~ w is a n ω - Z ∗ -lo cated w ord z ∈ e L (Σ , ~ k ) (resp. z ∈ L (Σ , ~ k )) with z = w n 1 ( p 1 , q 1 ) ⋆ . . . ⋆ w n λ ( p λ , q λ ) (resp. z = w n 1 ( p 1 ) ⋆ . . . ⋆ w n λ ( p λ )), where λ ∈ N , n 1 < . . . < n λ ∈ N and ( p i , q i ) ∈ { 1 , . . . , k n i } × { 1 , . . . , k − n i } (resp. p i ∈ { 1 , . . . , k n i } ) for ev ery 1 ≤ i ≤ λ . The set of all the extracted ω - Z ∗ -lo cated w ords of ~ w is denoted b y e E ( ~ w ) (resp. all the extracted ω -lo cated w ords of ~ w is denoted by E ( ~ w )). An extracted v ariable ω - Z ∗ -lo cated word (resp. extracted v ariabl e ω -lo cated w ord ) of ~ w is a v ariable ω - Z ∗ -lo cated w ord u ∈ e L 0 (Σ , ~ k ; υ ) (resp. u ∈ L (Σ , ~ k ; υ )) with u = w n 1 ( p 1 , q 1 ) ⋆ . . . ⋆ w n λ ( p λ , q λ ) (resp. u = w n 1 ( p 1 ) ⋆ . . . ⋆ w n λ ( p λ )), where λ ∈ N , n 1 < . . . < n λ ∈ N , ( p i , q i ) ∈ { 1 , . . . , k n i } × { 1 , . . . , k − n i } ∪ { ( υ , υ ) } for ev ery 1 ≤ i ≤ λ and ( υ , υ ) ∈ { ( p 1 , q 1 ) , . . . , ( p λ , q λ ) } (resp. p i ∈ { 1 , . . . , k n i } ∪ { υ } for ev ery 1 ≤ i ≤ λ and υ ∈ { p 1 , . . . , p λ } ). The set of all the extracted v ariable ω - Z ∗ -lo cated w or ds of ~ w is denoted by g E V ( ~ w ) (resp. the set of all the extracted v ariable ω -lo cated w or ds of ~ w is denoted by E V ( ~ w )). Let g E V ∞ ( ~ w ) = { ~ u = ( u n ) n ∈ N ∈ e L ∞ (Σ , ~ k ; υ ) : u n ∈ g E V ( ~ w ) for every n ∈ N } , E V ∞ ( ~ w ) = { ~ u = ( u n ) n ∈ N ∈ L ∞ (Σ , ~ k ; υ ) : u n ∈ E V ( ~ w ) for ev ery n ∈ N } . If ~ u ∈ g E V ∞ ( ~ w ) (resp. ~ u ∈ E V ∞ ( ~ w )), then w e sa y that ~ u is an extract ion of ~ w and w e write ~ u ≺ ~ w . Notice that fo r ~ u, ~ w ∈ e L ∞ (Σ , ~ k ; υ ) (resp. ~ u, ~ w ∈ L ∞ (Σ , ~ k ; υ )) w e hav e ~ u ≺ ~ w if and only if g E V ( ~ u ) ⊆ g E V ( ~ w ) (resp. E V ( ~ u ) ⊆ E V ( ~ w )). Using the theory of ultr a filters w e prov ed in [F], [FK] the follo wing partition theorem for ω - Z ∗ -lo cated w ords and for ω -lo cated w ords. Theorem 1.2. ( [F ] , [FK] ) L e t Σ = { α n : n ∈ Z ∗ } b e an al p h ab et, ~ k = ( k n ) n ∈ Z ∗ ⊆ N such t hat ( k n ) n ∈ N , ( k − n ) n ∈ N ar e incr e asing se quenc es, υ / ∈ Σ and let ~ w = ( w n ) n ∈ N ∈ e L ∞ (Σ , ~ k ; υ ) (r esp. ~ w = ( w n ) n ∈ N ∈ L ∞ (Σ , ~ k ; υ ) ). If e L (Σ , ~ k ) = C 1 ∪ . . . ∪ C s (r esp. L (Σ , ~ k ) = C 1 ∪ . . . ∪ C s ), s ∈ N , then ther e exists ~ u ≺ ~ w and 1 ≤ j 0 ≤ s such that e E ( ~ u ) ⊆ C j 0 (r esp. E ( ~ u ) ⊆ C j 0 ). 2. Implica tions of the p ar tition t heorem to topological dynamics W e will prov e a top ological form ula tion (in Theorem 2.1) of the partition The orem 1.2, imp ortan t f o r prov ing later (m ultiple) recurrence resu lts for systems of con tinu ous maps from a compact metric space into itself indexed by ω - Z ∗ -lo cated words (Theorem 2.6), 5 whic h e xtend fundamen tal recurrence res ults o f Birkhoff ([Bi]) a nd F ursten b erg-W eis s ([F u], [F uW]). Let an alphab et Σ = { α n : n ∈ Z ∗ } a nd ~ k = ( k n ) n ∈ Z ∗ ⊆ N , where ( k n ) n ∈ N , ( k − n ) n ∈ N are increasing sequences. Observ e that e L (Σ , ~ k ) can b e considered as a directed set with partial o rder either R 1 or R 2 . So, in a top ological space X , we can consider { x w } w ∈ e L (Σ , ~ k ) ⊆ X either as an R 1 -net or as an R 2 -net in X . Consequen tly , { x w } w ∈ L (Σ , ~ k ) is an R 2 -subnet of { x w } w ∈ e L (Σ , ~ k ) . Moreo v er, { x w } w ∈ e E ( ~ u ) for ~ u ∈ e L ∞ (Σ , ~ k ; υ ) is an R 1 -subnet of { x w } w ∈ e L (Σ , ~ k ) and respectiv ely { x w } w ∈ E ( ~ u ) for ~ u ∈ L ∞ (Σ , ~ k ; υ ) is an R 2 -subnet of { x w } w ∈ L (Σ , ~ k ) . Let x 0 ∈ X . W e write R 1 - lim w ∈ e L (Σ , ~ k ) x w = x 0 if { x w } w ∈ e L (Σ , ~ k ) con verges t o x 0 as R 1 -net in X , i.e. if for an y neigh b orho o d V of x 0 , there exists n 0 ≡ n 0 ( V ) ∈ N suc h that x w ∈ V for ev ery w with min {− max dom − ( w ) , min dom + ( w ) } ≥ n 0 . Analogously , we write R 2 - lim w ∈ L (Σ , ~ k ) x w = x 0 if f o r any neighborho o d V o f x 0 , there exists n 0 ≡ n 0 ( V ) ∈ N suc h that x w ∈ V f o r ev ery w with min dom ( w ) ≥ n 0 . W e will giv e no w a to p ological reform ulatio n of Theorem 1.2. Theorem 2.1. L et ( X , d ) b e a c omp act metric sp ac e, Σ = { α n : n ∈ Z ∗ } b e an alphab et, ~ k = ( k n ) n ∈ Z ∗ ⊆ N such that ( k n ) n ∈ N , ( k − n ) n ∈ N ar e incr e a s ing se quenc es , υ / ∈ Σ and ~ w = ( w n ) n ∈ N ∈ e L ∞ (Σ , ~ k ; υ ) (r esp. ~ w = ( w n ) n ∈ N ∈ L ∞ (Σ , ~ k ; υ ) ). F or ev ery net { x w } w ∈ e L (Σ , ~ k ) ⊆ X (r esp . { x w } w ∈ L (Σ , ~ k ) ⊆ X ), ther e exist an extr action ~ u ≺ ~ w of ~ w and x 0 ∈ X such that R 1 - lim w ∈ e E ( ~ u ) x w = x 0 ( r esp. R 2 - lim w ∈ E ( ~ u ) x w = x 0 ) . Pr o of. F or x ∈ X and ǫ > 0 we set b B ( x, ǫ ) = { y ∈ X : d ( x, y ) ≤ ǫ } . Since ( X , d ) is a compact metric space, w e hav e that X = S m 1 i =1 b B ( x 1 i , 1 2 ) for some x 1 1 , . . . , x 1 m 1 ∈ X . According to Theorem 1.2, there exists ~ u 1 ≺ ~ w and 1 ≤ i 1 ≤ m 1 suc h that { x w } w ∈ e E ( ~ u 1 ) ⊆ b B ( x 1 i 1 , 1 2 ) (resp. { x w } w ∈ E ( ~ u 1 ) ⊆ b B ( x 1 i 1 , 1 2 ) ). Analogously , since b B ( x 1 i 1 , 1 2 ) is compact, there exist x 2 1 , . . . , x 2 m 2 ∈ X , suc h that b B ( x 1 i 1 , 1 2 ) ⊆ S m 2 i =1 b B ( x 2 i , 1 4 ) , and conseque n tly there exist ~ u 2 ≺ ~ u 1 and 1 ≤ i 2 ≤ m 2 suc h that { x w } w ∈ e E ( ~ u 2 ) ⊆ b B ( x 1 i 1 , 1 2 ) ∩ b B ( x 2 i 2 , 1 4 ). Inductiv ely , we construct ( ~ u n ) n ∈ N ⊆ e L ∞ (Σ , ~ k ; υ ) (resp. ( ~ u n ) n ∈ N ⊆ L ∞ (Σ , ~ k ; υ )) suc h that ~ u n +1 ≺ ~ u n ≺ ~ w for ev ery n ∈ N and closed balls b B ( x n i n , 1 2 n ) , for n ∈ N suc h t ha t fo r eve ry n ∈ N 6 { x w } w ∈ e E ( ~ u n ) ⊆ T n j =1 b B ( x j i j , 1 2 j ) (resp. { x w } w ∈ E ( ~ u n ) ⊆ T n j =1 b B ( x j i j , 1 2 j )). If ~ u n = ( w ( n ) k ) k ∈ N for ev ery n ∈ N , then w e set ~ u = ( w ( n ) n ) n ∈ N . O f course ~ u ≺ ~ w . Let { x 0 } = T n ∈ N b B ( x n i n , 1 2 n ) . Then R 1 -lim w ∈ e E ( ~ u ) x w = x 0 (resp. R 2 -lim w ∈ E ( ~ u ) x w = x 0 ). Indeed, for ε > 0 pic k k 0 ∈ N suc h that 1 / 2 k 0 < ε. Then, for ev ery w ∈ e E ( ~ u k 0 ) w e ha v e that d ( x w , x 0 ) ≤ 1 / 2 k 0 < ε. Since e E ( ~ u n ) ⊆ e E ( ~ u k 0 ) for eve ry n ≥ k 0 , w e ha v e that e E (( w ( n ) n ) n ≥ k 0 ) ⊆ e E ( ~ u k 0 ) and consequen tly that { w ∈ e E ( ~ u ) : min {− max dom − ( w ) , min dom + ( w ) } ≥ n 0 } ⊆ e E ( ~ u k 0 ) for n 0 = max {− min dom − ( w ( k 0 ) k 0 ) , max dom + ( w ( k 0 ) k 0 ) } .  Remark 2.2. (1) Note that Theorem 2 .1 follo ws from The orem 1.2. But conv ersely Theorem 1.2 follo ws from Theorem 2 .1. In fa ct, one only needs t he assertion for finite spaces. Indeed, let e L (Σ , ~ k ) = C 1 ∪ . . . ∪ C s (resp. L (Σ , ~ k ) = C 1 ∪ . . . ∪ C s ), s ∈ N . Then defining, for ev ery w ∈ e L (Σ , ~ k ) (resp. fo r w ∈ L (Σ , ~ k )), x w = i if and only if w ∈ C i and w / ∈ C j for all j < i , w e ha v e, according to Theorem 2.1, that there exist ~ u = ( u n ) n ∈ N ≺ ~ w and 1 ≤ j 0 ≤ s such that R 1 - lim w ∈ e E ( ~ u ) x w = j 0 (resp. R 2 - lim w ∈ E ( ~ u ) x w = j 0 ) . F or n 0 large enough and ~ u 0 = ( u n + n 0 ) n ∈ N w e ha v e that e E ( ~ u 0 ) ⊆ C j 0 (resp. E ( ~ u 0 ) ⊆ C j 0 ). (2) Observ e that if R 1 - lim w ∈ e E ( ~ u ) x w = x 0 for ~ u = ( u n ) n ∈ N ∈ e L ∞ (Σ , ~ k ; υ ), then the sequence s ( x u n ( p n ,q n ) ) n ∈ N con verge uniformly to x 0 for a ll the sequences (( p n , q n )) n ∈ N ⊆ N × N with 1 ≤ p n ≤ k n , 1 ≤ q n ≤ k − n . Analogously , if R 2 - lim w ∈ E ( ~ u ) x w = x 0 for ~ u = ( u n ) n ∈ N ∈ L ∞ (Σ , ~ k ; υ ), then the seque nces ( x u n ( p n ) ) n ∈ N con verge uniformly to x 0 for all the sequences ( p n ) n ∈ N ⊆ N with 1 ≤ p n ≤ k n . (3) The particular case of Theorem 1.2 for w o r ds in L (Σ , ~ k ) , where Σ is a finite a lphab et, giv es Carlson’s partition theorem in [C], whose topo logical reformulation has been giv en b y F ursten b erg and Katznelson in [F uKa]. (4) The particular case of Theorem 1.2 for w ords in L (Σ , ~ k ) where Σ is a singleton and ~ k = ( k n ) n ∈ N with k n = 1 for a ll n ∈ N (so, the w ords c an be c oincide w ith its domain) is Hindman’s partition theorem in [H]. F urstenberg and W eiss in [F uW] ga v e the top ological refo r mulation of Hindman’s theorem in tro ducing the I P - con ve rgence of a net { x F } F ∈ [ N ] <ω > 0 in a top ological space X to x 0 ∈ X , i.e. if for an y neigh b orho o d V of x 0 , there exists n 0 ≡ n 0 ( V ) ∈ N suc h that x F ∈ V for ev ery F ∈ [ N ] <ω > 0 with min F ≥ n 0 . In this case w e write IP-lim F ∈ [ N ] <ω > 0 x F = x 0 . Also, using the I P -con vergenc e, they prov ed imp ortan t results in top o logical dynamics (see [F u]). In the f ollo wing prop osition w e will c haracterize the R 1 -con v erg ence of nets { x w } w ∈ e L (Σ , ~ k ) and the R 2 -con v erg ence of nets { x w } w ∈ L (Σ , ~ k ) as uniform I P -con v ergence, p oin ting out the w ay f o r strengthening results in volving the I P -con v ergence. 7 Prop osition 2.3. L et X b e a top olo gic al sp ac e, ~ w = ( w n ) n ∈ N ∈ e L ∞ (Σ , ~ k ; υ ) ( r es p . ~ w = ( w n ) n ∈ N ∈ L ∞ (Σ , ~ k ; υ )) and { x w } w ∈ e L (Σ , ~ k ) ⊆ X (r esp. { x w } w ∈ L (Σ , ~ k ) ⊆ X ). F or a se quenc e (( p n , q n )) n ∈ N ⊆ N × N w ith 1 ≤ p n ≤ k n , 1 ≤ q n ≤ k − n and F = { n 1 < . . . < n λ } ∈ [ N ] <ω > 0 a fi n ite non-empty subset of N w e set y (( p n ,q n )) n ∈ N F = x w n 1 ( p n 1 ,q n 1 ) ⋆...⋆w n λ ( p n λ ,q n λ ) (r esp. y ( p n ) n ∈ N F = x w n 1 ( p n 1 ) ⋆...⋆w n λ ( p n λ ) ). Then R 1 - lim w ∈ e E ( ~ w ) x w = x 0 if a n d only if I P - lim F ∈ [ N ] <ω > 0 y (( p n ,q n )) n ∈ N F = x 0 uniformly for al l se quenc es (( p n , q n )) n ∈ N ⊆ N × N with 1 ≤ p n ≤ k n , 1 ≤ q n ≤ k − n (r esp. R 2 - lim w ∈ E ( ~ w ) x w = x 0 if and only if I P - lim F ∈ [ N ] <ω > 0 y ( p n ) n ∈ N F = x 0 uniformly for al l se quenc es ( p n ) n ∈ N ⊆ N with 1 ≤ p n ≤ k n ). Pr o of. ( ⇒ ) L et V b e a neighbor ho o d of x 0 . There exists n 0 ≡ n 0 ( V ) ∈ N suc h that x w ∈ V for eve ry w ∈ e E ( ~ w ) (resp. w ∈ E ( ~ w ) ) with min {− max dom − ( w ) , min dom + ( w ) } ≥ n 0 (resp. with min d om ( w ) ≥ n 0 ). So, fo r F ∈ [ N ] <ω > 0 with n 0 < min F w e hav e that y (( p n ,q n )) n ∈ N F ∈ V (resp. y ( p n ) n ∈ N F ∈ V ) for all sequenc es (( p n , q n )) n ∈ N ⊆ N × N w ith 1 ≤ p n ≤ k n , 1 ≤ q n ≤ k − n (resp. ( p n ) n ∈ N ⊆ N with 1 ≤ p n ≤ k n ). ( ⇐ ) T ow ard to a con tradiction w e supp ose that there exists a neighborho o d V of x 0 suc h that for ev ery n ∈ N there exists u n = w m n 1 ( p m n 1 , q m n 1 ) ⋆ . . . ⋆ w m n λ ( p m n λ , q m n λ ) ∈ e E ( ~ w ) (resp. u n = w m n 1 ( p m n 1 ) ⋆ . . . ⋆ w m n λ ( p m n λ ) ∈ E ( ~ w )) with min {− max dom − ( u n ) , min dom + ( u n ) } ≥ n (resp. min dom ( u n ) ≥ n ) and x u n / ∈ V . W e can supp ose that u n < R 1 u n +1 (resp. u n < R 2 u n +1 ) for ev ery n ∈ N . According to the h yp othesis there exists n 0 ∈ N suc h that y (( p n ,q n )) n ∈ N F ∈ V (resp. y ( p n ) n ∈ N F ∈ V ) fo r all sequences (( p n , q n )) n ∈ N ⊆ N × N w ith 1 ≤ p n ≤ k n , 1 ≤ q n ≤ k − n (resp. ( p n ) n ∈ N ⊆ N with 1 ≤ p n ≤ k n ) and all F ∈ [ N ] <ω > 0 with min F ≥ n 0 . Then x u n 0 ∈ V , a con tra diction.  W e will no w give some applications of Theorem 2.1 to top ological dynamical sys- tems extending fundamen tal recurrence results of Birkhoff ([Bi ]) and F urste n b erg-W eis s ([F uW], [F u]). Firstly , w e will intro duce the notions of e L (Σ , ~ k )- system s and L (Σ , ~ k )- systems of con tinu ous maps of a top ological space into itself. Definition 2.4. Let X b e a top ological space, Σ = { α n : n ∈ Z ∗ } b e an alphab et and ~ k = ( k n ) n ∈ Z ∗ ⊆ N suc h that ( k n ) n ∈ N , ( k − n ) n ∈ N are increasing sequences. A family { T w } w ∈ e L (Σ , ~ k ) (resp. { T w } w ∈ L (Σ , ~ k ) ) of con tin uous functions of X in to itself is an e L (Σ , ~ k ) - system ( r esp. an L (Σ , ~ k ) -system ) of X if T w 1 T w 2 = T w 1 ⋆w 2 for w 1 < R 1 w 2 (resp. for w 1 < R 2 w 2 ). Example 2.5. Let X b e a top ological space. 8 (1) Let T : X → X be a con tinuous map. F o r an alphab et Σ = ( m n ) n ∈ N ⊆ N , ~ k = ( k n ) n ∈ N ⊆ N an inc reasing sequence and ( l n ) n ∈ N ⊆ N w e define for eve ry w = w n 1 . . . w n λ ∈ L (Σ , ~ k ) T w = T l n 1 w n 1 + ... + l n λ w n λ . Then { T w } w ∈ L (Σ , ~ k ) is an L (Σ , ~ k )-system of X . Moreo v er, for a sequence { T n } n ∈ N of con tinuous maps fro m X in to itself defining T w = T l n 1 w n 1 n 1 . . . T l n λ w n λ n λ . w e hav e another L (Σ , ~ k )-system of X . (2) F or a giv en sequenc e { T n } n ∈ Z ∗ of con tin uous maps fro m X in to itself, Σ = ( α n ) n ∈ Z ∗ ⊆ N , ~ k = ( k n ) n ∈ Z ∗ ⊆ N suc h that ( k n ) n ∈ N , ( k − n ) n ∈ N are increasing sequences and ( l n ) n ∈ Z ∗ ⊆ N w e define for w = w n 1 . . . w n λ ∈ e L (Σ , ~ k ) T w n 1 ...w n λ = T l n 1 w n 1 n 1 . . . T l n λ w n λ n λ . Then { T w } w ∈ e L (Σ , ~ k ) is an e L (Σ , ~ k )-system of X . In particular, if T , S : X → X are t w o contin uous maps, then we can replace T n with T n and T − n with S n for ev ery n ∈ N . Via Theorem 2.1, w e will pro v e the existence of strongly r ecurrent p oints in a compact metric space X for an e L (Σ , ~ k )- system as we ll as for an L (Σ , ~ k )- system of it. Moreov er, w e will p oin t out the w ay t o lo cate such points. Theorem 2.6. L et { T w } w ∈ e L (Σ , ~ k ) (r esp. { T w } w ∈ L (Σ , ~ k ) ) b e an e L (Σ , ~ k ) -system (r esp. L (Σ , ~ k ) - system) of a c omp act metric sp ac e ( X , d ) , ~ w ∈ e L ∞ (Σ , ~ k ; υ ) (r esp. ~ w ∈ L ∞ (Σ , ~ k ; υ ) ) and x ∈ X . Then ther e ex ist an extr action ~ u ≺ ~ w of ~ w and x 0 ∈ X such that R 1 - lim w ∈ e E ( ~ u ) T w ( x ) = x 0 ( r esp. R 2 - lim w ∈ E ( ~ u ) T w ( x ) = x 0 ) . Mor e over, x 0 is ~ w -r e curr ent p oint , in the se n se that R 1 - lim w ∈ e E ( ~ u ) T w ( x 0 ) = x 0 ( r esp. R 2 - lim w ∈ E ( ~ u ) T w ( x 0 ) = x 0 ) . Pr o of. According to The orem 2.1 there exist an extraction ~ u of ~ w and x 0 ∈ X suc h that R 1 - lim w ∈ e E ( ~ u ) T w ( x ) = x 0 (resp. R 2 - lim w ∈ E ( ~ u ) T w ( x ) = x 0 ) . Let ǫ > 0. There e xists n 0 ∈ N such t hat d ( T w ( x ) , x 0 ) < ǫ 2 for ev ery w ∈ e E ( ~ u ) with min {− max dom − ( w ) , min dom + ( w ) } ≥ n 0 (resp. w ∈ E ( ~ u ) with min d om ( w ) ≥ n 0 ). Let w ∈ e E ( ~ u ) with min {− max dom − ( w ) , min dom + ( w ) } ≥ n 0 (resp. w ∈ E ( ~ u ) with min d om ( w ) ≥ n 0 ). Then d ( T w ( x ) , x 0 ) < ǫ 2 . Since T w is contin uous, there exists δ > 0 suc h that if d ( z , x 0 ) < δ , then d ( T w ( z ) , T w ( x 0 )) < ǫ 2 . Cho ose w 1 ∈ e E ( ~ u ) (resp. 9 w 1 ∈ E ( ~ u )) suc h that d ( T w 1 ( x ) , x 0 ) < δ and w < R 1 w 1 (resp. w < R 2 w 1 ). The n d ( T w ( T w 1 ( x )) , T w ( x 0 )) = d ( T w ⋆w 1 ( x ) , T w ( x 0 )) < ǫ 2 . Since d ( T w ⋆w 1 ( x ) , x 0 ) < ǫ 2 w e ha ve that d ( T w ( x 0 ) , x 0 ) < ǫ .  In t he follow ing coro llaries we will describe some consequences of Theorem 2.6 f o r the simplest e L (Σ , ~ k )- system g enerated b y a single transformation. F or a semigroup ( X , +) and ( x n ) n ∈ N ⊆ X let F S (( x n ) n ∈ N ) = { x n 1 + . . . + x n λ : λ ∈ N , n 1 < . . . < n λ ∈ N } . Corollary 2.7. L et ( X , d ) b e a c omp act me tric sp ac e, T : X → X a c ontinuous map and ( m n ) n ∈ N , ( r n ) n ∈ N ⊆ N wi th m n < m n +1 , r n < r n +1 for n ∈ N . Then, ther e exist x 0 ∈ X and se quen c es ( α n ) n ∈ N ⊆ N , ( β n ) n ∈ N ⊆ F S (( m n ) n ∈ N ) , ( γ n ) n ∈ N ⊆ F S (( r n ) n ∈ N ) such that IP- lim F ∈ [ N ] <ω > 0 T P n ∈ F α n + p n β n + q n γ n ( x 0 ) = x 0 , ( in p articular , lim n T α n + p n β n + q n γ n ( x 0 ) = x 0 ) uniformly for al l se q uenc es (( p n , q n )) n ∈ N ⊆ N × N with 0 ≤ p n ≤ n , 0 ≤ q n ≤ n . Pr o of. Let Σ = ( α n ) n ∈ Z ∗ ⊆ N with α − n = α n = n for n ∈ N and ~ k = ( k n ) n ∈ Z ∗ ⊆ N with k − n = k n = n + 1 for n ∈ N . F or w = w n 1 . . . w n λ ∈ e L (Σ , ~ k ) w e set T w n 1 ...w n λ = T − n 1 w n 1 . . . T − n i w n i T n i +1 w n i +1 . . . T n λ w n λ , whe re n i = max d om − ( w ) , n i +1 = min dom + ( w ) . Then { T w } w ∈ e L (Σ , ~ k ) is an e L (Σ , ~ k )-system of X (see Example 2.5(2)). Let ~ w = ( w n ) n ∈ N ∈ e L ∞ (Σ , ~ k ; υ ) with w n = w − r n w m n where w − r n = w m n = υ . W e apply T heorem 2 .6. So, the re exist an extraction ~ u = ( u n ) n ∈ N ∈ e L ∞ (Σ , ~ k ; υ ) of ~ w and x 0 ∈ X suc h that R 1 - lim w ∈ e E ( ~ u ) T w ( x 0 ) = x 0 . According to Proposition 2.3, if y (( p n ,q n )) n ∈ N F = T u n 1 ( p n 1 ,q n 1 ) ⋆...⋆u n λ ( p n λ ,q n λ ) ( x 0 ), then I P -lim F ∈ [ N ] <ω > 0 y (( p n ,q n )) n ∈ N F = x 0 uniformly for all seq uences (( p n , q n )) n ∈ N ⊆ N × N with 1 ≤ p n ≤ n + 1, 1 ≤ q n ≤ n + 1. Let T u n (( p n ,q n )) = T α n +( p n − 1) β n +( q n − 1) γ n , where β n ∈ F S (( m n ) n ∈ N ) and γ n ∈ F S (( r n ) n ∈ N ). Then I P -lim F ∈ [ N ] <ω > 0 T P n ∈ F α n + p n β n + q n γ n ( x 0 ) = x 0 , (in particular, lim T α n + p n β n + q n γ n ( x 0 ) = x 0 ) uniformly for all seq uences (( p n , q n )) n ∈ N ⊆ N × N with 0 ≤ p n ≤ n , 0 ≤ q n ≤ n .  Corollary 2.8. L et ( X, d ) b e a c om p act metric sp ac e, T : X → X a c ontinuous map and ( m n ) n ∈ N , ( r n ) n ∈ N ⊆ N with m n < m n +1 , r n < r n +1 for al l n ∈ N . Then, ther e exist x 0 ∈ X and se quenc es ( α n ) n ∈ N ⊆ N , ( β n ) n ∈ N ⊆ F S (( m n ) n ∈ N ) , ( γ n ) n ∈ N ⊆ F S (( r n ) n ∈ N ) such that for ev ery ǫ > 0 ther e exists n 0 ∈ N which satisfies d ( T p n β n + q n γ n ( T α n ( x 0 )) , T α n ( x 0 )) < ǫ 10 for every n ≥ n 0 and (( p n , q n )) n ∈ N ⊆ N × N with 0 ≤ p n ≤ n , 0 ≤ q n ≤ n . Pr o of. It follo ws from Corollary 2.7.  W e will define now the recurren t subsets of a compact metric space X for an e L (Σ , ~ k )- system as w ell as for an L (Σ , ~ k )- system o f it . Definition 2.9. Let { T w } w ∈ e L (Σ , ~ k ) (resp. { T w } w ∈ L (Σ , ~ k ) ) b e an e L (Σ , ~ k )- system (resp. L (Σ , ~ k )-system) of con tinuous maps of a compact metric space ( X , d ) a nd ~ w ∈ e L ∞ (Σ , ~ k ; υ ) (resp. ~ w ∈ L ∞ (Σ , ~ k ; υ )). A closed subset A of X is said to b e ~ w -recurr en t set if for an y m ∈ N , ε > 0 and an y p o in t x ∈ A there ex ist y ∈ A and u ∈ g E V ( ~ w ) with min {− max d om − ( u ) , min dom + ( u ) } > m (resp. u ∈ E V ( ~ w ) with min dom ( u ) } > m ) suc h t ha t d ( T u ( p,q ) ( y ) , x ) < ε for ev ery 1 ≤ p, q ≤ m. In the follo wing example w e p oin t out the w a y to lo cate recurren t subsets of a compact metric space X for a giv en e L (Σ , ~ k )- system a s w ell as for a giv en L (Σ , ~ k )- system o f it . Example 2.10. Let ( X , d ) b e a compact metric space and let F ( X ) b e the set of all nonempt y closed subsets of X endo w ed with the Ha usdorff metric ˆ d (where ˆ d ( A, B ) = max[sup x ∈ A d ( x, B ) , sup x ∈ B d ( x, A )]). Then ( F ( X ) , ˆ d ) is also a compact metric space. Let { T w } w ∈ e L (Σ , ~ k ) (resp. { T w } w ∈ L (Σ , ~ k ) ) b e an e L (Σ , ~ k )-system (r esp. L (Σ , ~ k )-system) of contin uous maps of ( X , d ). W e define ˆ T w : F ( X ) → F ( X ) with ˆ T w ( A ) = T w ( A ). Then { ˆ T w } w ∈ e L (Σ , ~ k ) (resp. { ˆ T w } w ∈ L (Σ , ~ k ) ) is an e L (Σ , ~ k )-system (resp. L (Σ , ~ k )-system) of ( F ( X ) , ˆ d ). According to Theorem 2.6, for ev ery ~ w = ( w n ) n ∈ N ∈ e L ∞ (Σ , ~ k ; υ ) (resp. ~ w = ( w n ) n ∈ N ∈ L ∞ (Σ , ~ k ; υ )) there exist A ∈ F ( X ) a nd an extraction ~ u ≺ ~ w of ~ w suc h that R 1 - lim w ∈ e E ( ~ u ) ˆ T w ( A ) = A (resp. R 2 - lim w ∈ E ( ~ u ) ˆ T w ( A ) = A ) . Then A is ~ w -recurren t in ( X , d ). Observ e that it is enough R 1 - lim w ∈ e E ( ~ u ) ˆ T w ( A ) ⊇ A (resp. R 2 - lim w ∈ E ( ~ u ) ˆ T w ( A ) ⊇ A ) in order A to b e ~ w -recurren t. Prop osition 2.11. L et A b e a ~ w -r e curr ent s ubse t of a c omp act metric sp ac e ( X , d ) . Then fo r every ε > 0 and m ∈ N t her e exist u ∈ g E V ( ~ w ) with min {− max dom − ( u ) , min dom + ( u ) } > m ( r esp. u ∈ E V ( ~ w ) with min dom ( u ) > m ) an d z ∈ A such that d ( T u ( p,q ) ( z ) , z ) < ε for every 1 ≤ p, q ≤ m. Pr o of. Let ε > 0 and m ∈ N . F or a z 0 ∈ A and ε 1 = ε/ 2 there exis t z 1 ∈ A and u 1 ∈ g E V ( ~ w ) with min {− max dom − ( u 1 ) , min dom + ( u 1 ) } > m ( r esp. u 1 ∈ E V ( ~ w ) with min dom ( u ) > m ) suc h that d ( T u 1 ( p,q ) z 1 , z 0 ) < ε for ev ery 1 ≤ p, q ≤ m. 11 Let ha ve b een c hosen z 0 , z 1 , . . . , z r ∈ A, u 1 < R 1 . . . < R 1 u r ∈ g E V ( ~ w ) (resp. u 1 < R 2 . . . < R 2 u r ∈ E V ( ~ w )) suc h that d ( T u i ( p i ,q i ) ⋆...⋆u j ( p j ,q j ) ( z j ) , z i − 1 ) < ε/ 2 f or ev ery 1 ≤ i ≤ j ≤ r and 1 ≤ p l , q l ≤ m, for all i ≤ l ≤ j. Since T w are con tinuous functions, there is ε r < ε/ 2 suc h that if d ( z , z r ) < ε r then d ( T u i ( p i ,q i ) ⋆...⋆u r ( p r ,q r ) ( z ) , z i − 1 ) < ε / 2 fo r ev ery 1 ≤ i ≤ r and 1 ≤ p l , q l ≤ m, for all i ≤ l ≤ r. Since A is ~ w -recurren t, there exist z r +1 ∈ A and u r +1 ∈ g E V ( ~ w ) with u r < R 1 u r +1 (resp. u r +1 ∈ E V ( ~ w ) with u r < R 2 u r +1 ) suc h that d ( T u r +1 ( p,q ) ( z r +1 ) , z r ) < ε r for ev ery 1 ≤ p, q ≤ m. Hence, d ( T u i ( p i ,q i ) ⋆...⋆u r +1 ( p r +1 ,q r +1 ) ( z r +1 ) , z i − 1 ) < ε/ 2 for ev ery 1 ≤ i ≤ r + 1 and 1 ≤ p l , q l ≤ m, for all i ≤ l ≤ r + 1 . Since ( X , d ) is compact, there exist i < j ∈ N such that d ( z i , z j ) < ε/ 2. Hence, for u = u i +1 ⋆ . . . ⋆ u j ∈ g E V ( ~ w ) (resp. u = u i +1 ⋆ . . . ⋆ u j ∈ E V ( ~ w )) w e hav e d ( T u ( p,q ) z j , z j ) < ε for ev ery 1 ≤ p, q ≤ m.  Definition 2.12. A closed subset A of a compact metric space X is homogeneous with res p ect to a set of transfor ma t ions { T i } acting on X if there exists a group of homeomorphisms G of X eac h of whic h comm utes with each T i and such that G leav es A in v arian t a nd ( A, G ) is minimal (no prop er closed subset o f A is in v arian t under the action of G ). In the follo wing prop osition w e giv e a sufficien t condition in or der a homogeneous subset to be strongly recurren t . Prop osition 2.13. L et A is a homo gene ous set in a c omp act metric s p ac e X with r esp e ct to the system { T w } w ∈ e L (Σ , ~ k ) (r esp. { T w } w ∈ L (Σ , ~ k ) ) a nd ~ w ∈ e L ∞ (Σ , ~ k ; υ ) ( r esp. ~ w ∈ L ∞ (Σ , ~ k ; υ ) ). If for every ε > 0 and m ∈ N ther e exist x, y ∈ A and u ∈ g E V ( ~ w ) with min {− max d om − ( u ) , min dom + ( u ) } > m (r esp. u ∈ E V ( ~ w ) with m in d om ( u ) } > m ) such that d ( T u ( p,q ) ( y ) , x ) < ε for every 1 ≤ p, q ≤ m, then A is ~ w -r e curr ent. Pr o of. Let ε > 0, m ∈ N , and G be a group of homeomorphisms comm uting with { T w } , and such that G leav es A in v arian t and ( A, G ) is minimal. Let { U 1 , . . . , U r } b e a finite co v ering of A by op en sets of diameter < ε/ 2 . Then, from the m inimalit y of A , w e can find for eac h 1 ≤ i ≤ r a finite set { g i 1 , . . . , g l i 1 } ⊆ G suc h t ha t S l i j =1 ( g i j ) − 1 ( U i ) = A. Let G 0 = { g i j : 1 ≤ i ≤ r , 1 ≤ j ≤ l i } ⊆ G . Then for an y x, y ∈ A w e ha v e min g ∈ G 0 d ( g ( x ) , y ) < ε/ 2 . Let δ > 0 suc h tha t if d ( x 1 , x 2 ) < δ , then d ( g ( x 1 ) , g ( x 2 )) < ε for ev ery g ∈ G 0 . Ac- cording to the h yp othesis, there exis t x , y ∈ A and u ∈ g E V ( ~ w ) with min {− max dom − ( u ) , min dom + ( u ) } > m (resp. u ∈ E V ( ~ w ) with min dom ( u ) > m ) suc h that d ( T u ( p,q ) ( y ) , x ) < δ for ev ery 1 ≤ p, q ≤ m . Then 12 d ( T u ( p,q ) ( g ( y )) , g ( x )) = d ( g ( T u ( p,q ) ( y )) , g ( x )) < ε/ 2 for ev ery 1 ≤ p, q ≤ m. F or a p oint z ∈ A, find g ∈ G 0 with d ( g ( x ) , z ) < ε/ 2. Then d ( T u ( p,q ) ( g ( y )) , z ) ≤ d ( T u ( p,q ) ( g ( y )) , g ( x )) + d ( g ( x ) , z ) < ε for ev ery 1 ≤ p, q ≤ m. It follo ws tha t A is ~ w - recurren t.  W e will prov e no w that a recurren t homogeneous subset A of a compact metric space X con tains recurrent po ints, mor eov er these p oints consist a dense subset of A. Prop osition 2.14. L et { T w } w ∈ e L (Σ , ~ k ) (r esp. { T w } w ∈ L (Σ , ~ k ) ) b e a n e L (Σ , ~ k ) -system (r esp. L (Σ , ~ k ) -system) of c ontinuous tr ansfo rm ations o f a c om p act m e tric sp ac e ( X , d ) and ~ w ∈ e L ∞ (Σ , ~ k ; υ ) ( r esp. ~ w ∈ L ∞ (Σ , ~ k ; υ ) ). A ~ w -r e curr ent homo gene ous subset A of X c ontain s ~ w -r e curr ent p oints ( x 0 is ~ w -r e curr ent iff R 1 - lim w ∈ e E ( ~ u ) T w ( x 0 ) = x 0 (r esp. iff R 2 - lim w ∈ E ( ~ u ) T w ( x 0 ) = x 0 ) fo r some ~ u ≺ ~ w ). Mor e over, the ~ w -r e curr ent p oints of A c onsist a dense subset of A . Pr o of. Let V b e an op en subset of X suc h that V ∩ A 6 = ∅ and let V ′ ⊆ V b e an op en set suc h t ha t V ′ ∩ A 6 = ∅ and if d ( x, V ′ ) < δ for δ > 0 then x ∈ V . Since A is homog eneous, there exists a group G of homeomorphisms comm uting with { T w } and suc h that G lea ves A inv arian t and ( A, G ) is minimal. F ro m the minimalit y of A , there exists a finite subset G 0 ⊆ G suc h that A ⊆ S g ∈ G 0 g − 1 ( V ′ ). Cho ose ε > 0 suc h that whenev er x 1 , x 2 ∈ X and d ( x 1 , x 2 ) < ε, then d ( g ( x 1 ) , g ( x 2 )) < δ for ev ery g ∈ G 0 . Since A is ~ w -recurren t, according to Prop osition 2.11, for m ∈ N there exist z ∈ A and u ∈ g E V ( ~ w ) with min {− max dom − ( u ) , min dom + ( u ) } > m (resp. u ∈ E V ( ~ w ) with min dom ( u ) > m ) suc h that d ( T u ( p,q ) ( z ) , z ) < ε for all 1 ≤ p, q ≤ m. There exis ts g ∈ G 0 with g ( z ) ∈ V ′ and since d ( T u ( p,q ) ( g ( z )) , g ( z ) ) < δ for ev ery 1 ≤ p, q ≤ m, w e ha v e that T u ( p,q ) ( g ( z )) ∈ V fo r ev ery 1 ≤ p, q ≤ m. Hence , eac h op en set V with V ∩ A 6 = ∅ con ta ins a p oint z ′ = g ( z ) ∈ A with T u ( p,q ) z ′ ∈ V for ev ery 1 ≤ p, q ≤ m. Since T w are con tinuous, w e conclude that f or ev ery op en set V with V ∩ A 6 = ∅ and ev ery m ∈ N there exists an op en set V 1 suc h that V 1 ⊆ V and T u ( p,q ) V 1 ⊆ V for ev ery 1 ≤ p, q ≤ m, for some u ∈ g E V ( ~ w ) with min {− max dom − ( u ) , min dom + ( u ) } > m (resp. u ∈ E V ( ~ w ) with min dom ( u ) > m ). Let V 0 b e an op en subset of X suc h that V 0 ∩ A 6 = ∅ . Inductiv ely w e can define a sequence ( V n ) n ∈ N of op en sets and a sequence ~ u = ( u n ) n ∈ N ∈ e L ∞ (Σ , ~ k ; υ ) (resp. ~ u = ( u n ) n ∈ N ∈ L ∞ (Σ , ~ k ; υ )) with ~ u ≺ ~ w suc h that V n ⊆ V n − 1 , V n ∩ A 6 = ∅ and T u n ( p n ,q n ) V n ⊆ V n − 1 for ev ery n ∈ N and 1 ≤ p n ≤ k n , 1 ≤ q n ≤ k − n . W e can also supp ose that t he diameter of V n tends to 0. The n T n ∈ N V n ∩ A = { x 0 } . 13 F or 1 < i 1 < . . . < i k , w e ha v e that T u i 1 ( p i 1 ,q i 1 ) ⋆...⋆u i k ( p i k ,q i k ) V i k ⊆ V i 1 − 1 . Then T w ( x 0 ) ∈ V i for every w ∈ e E ( ~ u ) with u i +1 < R 1 w (resp. w ∈ E ( ~ u ) with u i +1 < R 2 w ) so R 1 - lim w ∈ e E ( ~ u ) T w ( x 0 ) = x 0 (resp. R 2 -lim w ∈ e E ( ~ u ) T w ( x 0 ) = x 0 ). Hence, x 0 ∈ A ∩ V 0 is a ~ w - recurren t point. This gives that the set of ~ w -recurren t p o ints in A is dense in A .  No w, w e shall prov e a multiple recurrence theorem extending Theorem 2.6, in case the tr ansformations are homeomorphisms. W e can sa y that the follow ing theorem is the “w o rd”-analogue of Birkhoff ’s m ultiple recurrence theorem. Theorem 2.15. L et { T w 1 } w ∈ e L (Σ , ~ k ) , . . . , { T w m } w ∈ e L (Σ , ~ k ) ( r esp. { T w 1 } w ∈ L (Σ , ~ k ) , . . . , { T w m } w ∈ L (Σ , ~ k ) ) b e m systems of tr ansf ormations o f a c om p act metric sp ac e X , a l l c ontaine d in a c om mu- tative gr oup G of home omorphism s of X and let ~ w ∈ e L ∞ (Σ , ~ k ; υ ) (r esp. ~ w ∈ L ∞ (Σ , ~ k ; υ ) ). Then, ther e exist x 0 ∈ X and an extr action ~ u ≺ ~ w such that R 1 - lim w ∈ e E ( ~ u ) T w i ( x 0 ) = x 0 (r esp. R 2 - lim w ∈ E ( ~ u ) T w i ( x 0 ) = x 0 ) for every 1 ≤ i ≤ m. Mor e over, in c a s e ( X , G ) is m inimal, the set of such p oints x 0 is a dense subset of X . Pr o of. W e a ssume without loss of generality that ( X , G ) is minimal, otherwise we re- place X b y a G -minimal subset of X . F or m = 1 we obtain the assertion from The- orem 2.6. W e pro ceed b y induction. Supp ose that the theorem holds for m ∈ N and that { T w 1 } w ∈ e L (Σ , ~ k ) , . . . , { T w m +1 } w ∈ e L (Σ , ~ k ) (resp. { T w 1 } w ∈ L (Σ , ~ k ) , . . . , { T w m +1 } w ∈ L (Σ , ~ k ) ) are m + 1 suc h systems. W e set S w i = T w i ( T w m +1 ) − 1 for all 1 ≤ i ≤ m. Then S w 1 ⋆w 2 i = S w 1 i S w 2 i for ev ery 1 ≤ i ≤ m a nd w 1 < R 1 w 2 (resp. w 1 < R 2 w 2 ), since all the maps commu te. By the induction hy p othesis there exist y ∈ X and ~ u ≺ ~ w suc h that R 1 -lim w ∈ e E ( ~ u ) S w i ( y ) = y (resp. R 2 -lim w ∈ E ( ~ u ) S w i ( y ) = y ) for ev ery 1 ≤ i ≤ m. Consider the pro duct X m +1 and let ∆ m +1 b e the diago nal subset cons isting of the ( m + 1)-tuples ( x, . . . , x ) ∈ X m +1 . Identifyin g eac h g ∈ G with g × . . . × g w e can assume that G acts on X m +1 . Also, the functions T w 1 × . . . × T w m +1 acts on X m +1 and comm ute with the functions o f G . Since G lea v es ∆ m +1 in v arian t and (∆ m +1 , G ) is minimal, ∆ m +1 is a homogeneous set. According to Pro p osition 2.1 4, it suffices to pro v e that ∆ m +1 is ~ w -recurren t. But, according to Prop o sition 2.1 3, the set ∆ m +1 is ~ w - recurren t, since R 1 - lim w ∈ e E ( ~ u ) ( T w 1 × . . . × T w m +1 )[(( T w m +1 ) − 1 × . . . × ( T w m +1 ) − 1 )(( y , . . . , y ))] = ( y , . . . , y ) (resp. R 2 - lim w ∈ E ( ~ u ) ( T w 1 × . . . × T w m +1 )[(( T w m +1 ) − 1 × . . . × ( T w m +1 ) − 1 )(( y , . . . , y ))] = ( y , . . . , y )).  Theorem 2.15 has the fo llo wing consequence. 14 Prop osition 2.16. L et { T w 1 } w ∈ e L (Σ , ~ k ) , . . . , { T w m } w ∈ e L (Σ , ~ k ) ( r esp. { T w 1 } w ∈ L (Σ , ~ k ) , . . . , { T w m } w ∈ L (Σ , ~ k ) ) b e m systems of tr ansformations of a c om p act metric sp ac e X , al l c ontaine d in a c ommuta- tive gr oup G o f home omorphisms of X , which acts minim al l y on X . F or ~ w ∈ e L ∞ (Σ , ~ k ; υ ) (r esp. ~ w ∈ L ∞ (Σ , ~ k ; υ ) ) and U a non-empty op e n subset of X , ther e exists ~ u ≺ ~ w so that m \ i =1 ( T w i ) − 1 ( U ) 6 = ∅ for every w ∈ e E ( ~ u ) ( r esp. w ∈ E ( ~ u )) . Pr o of. Since G acts minimally on X , X = S g ∈ G 0 g − 1 ( U ), where G 0 is a finite su bset of G . Let δ > 0 b e suc h that ev ery set of diameter < δ is contained in some g − 1 ( U ) for g ∈ G 0 . According to Theorem 2.15, t here exist x 0 ∈ X and ~ u ≺ ~ w such that R 1 - lim w ∈ e E ( ~ u ) T w i ( x 0 ) = x 0 (resp. R 2 -lim w ∈ E ( ~ u ) T w i ( x 0 ) = x 0 ) for ev ery 1 ≤ i ≤ m . Refine ~ u suc h that d ( T w i ( x 0 ) , x 0 ) < δ / 2 for ev ery w ∈ e E ( ~ u ) (resp. w ∈ E ( ~ u )) and 1 ≤ i ≤ m. Then there exists g ∈ G 0 suc h that T w i ( x 0 ) ∈ g − 1 ( U ) for ev ery w ∈ e E ( ~ u ) (resp. w ∈ E ( ~ u )) a nd 1 ≤ i ≤ m . Consequen tly , g ( x 0 ) ∈ T m i =1 ( T w i ) − 1 ( U ) for ev ery w ∈ e E ( ~ u ) (resp. w ∈ E ( ~ u ) ) .  3. Applica tions W e will indicate the wa y in whic h the recurrence res ults for to p ological systems or nets indexed b y w ords, that w e prov ed in t he previous section, can b e applied to systems or nets indexed b y semigroups that can b e represen t ed as w ords (Example 1 .1 ) and consequen tly to systems or nets indexed b y an arbitrary semigroup. Semigroup ( Q , +) . As we described in Example 1 .1 (1), the set Q ∗ of the nonzero ra t io- nal num b ers can be iden t ified with a set e L (Σ , ~ k ) of ω - Z ∗ -lo cated w ords, via the function g : e L (Σ , ~ k ) → Q ∗ , with g ( q t 1 . . . q t l ) = P t ∈ dom − ( w ) q t ( − 1) − t ( − t +1)! + P t ∈ dom + ( w ) q t ( − 1) t +1 t ! . W e extend the function g to the set e L (Σ , ~ k ; υ ) of v ariable words corresp onding to eac h w = q t 1 . . . q t l ∈ e L (Σ , ~ k ; υ ) a function q = g ( w ) which sends every ( i, j ) ∈ N × N with 1 ≤ i ≤ − ma x dom − ( w ) , 1 ≤ j ≤ min d o m + ( w ) , to q ( i, j ) = g ( T ( j,i ) ( w )) = X t ∈ C − q t ( − 1) − t ( − t + 1)! + i X t ∈ V − ( − 1) − t ( − t + 1)! + X t ∈ C + q t ( − 1) t +1 t !+ j X t ∈ V + ( − 1) t +1 t ! , where C − = { t ∈ dom − ( w ) : q t ∈ Σ } , V − = { t ∈ dom − ( w ) : q t = υ } and C + = { t ∈ dom + ( w ) : q t ∈ Σ } , V + = { t ∈ dom + ( w ) : q t = υ } . Let Q ( υ ) = g ( e L (Σ , ~ k ; υ )). T hen the extended function g : e L (Σ ∪ { υ } , ~ k ) → Q ∗ ∪ Q ( υ ) 15 is one-to-one and onto. F or q 1 , q 2 ∈ Q ∗ ∪ Q ( υ ) w e define the relation q 1 < R 1 q 2 ⇐ ⇒ g − 1 ( q 1 ) < R 1 g − 1 ( q 2 ) . So, { x q } q ∈ Q ∗ ⊆ X , where X is a top ological space, can b e considered as an R 1 -net and consequen tly w e can define, for x 0 ∈ X , R 1 -lim q ∈ Q ∗ x q = x 0 iff for any neigh b o r- ho o d V of x 0 , there exists n 0 ≡ n 0 ( V ) ∈ N suc h tha t x q ∈ V fo r ev ery q ∈ Q ∗ with min {− max d om − ( g − 1 ( q )) , min dom + ( g − 1 ( q )) } ≥ n 0 . Observ e that g ( w 1 ⋆ w 2 ) = g ( w 1 ) + g ( w 2 ) for eve ry w 1 < R 1 w 2 ∈ e L (Σ ∪ { υ } , ~ k ) . So , if ~ q = ( q n ) n ∈ N ∈ Q ∞ ( υ ) = { ( q n ) n ∈ N : q n ∈ Q ( υ ) and q n < R 1 q n +1 } , then the set of the extractions of ~ q is g E V ∞ ( ~ q ) = { ~ r = ( r n ) n ∈ N ∈ Q ∞ ( υ ) : r n = g ( u n ) for ( u n ) n ∈ N ∈ g E V ∞ (( g − 1 ( q n )) n ∈ N ) } a nd the set o f a ll the extracted rationa ls of ~ q is e E ( ~ q ) = { q ∈ F S [( q n ( i n , j n )) n ∈ N ] : (( i n , j n )) n ∈ N ⊆ N × N with 1 ≤ i n , j n ≤ n } = = { g ( w ) : w ∈ e E (( g − 1 ( q n )) n ∈ N ) } . Of course, { x q } q ∈ e E ( ~ q ) is an R 1 -subnet of { x q } q ∈ Q ∗ . Hence, via the function g , all the presen ted results relating to ω - Z ∗ -lo cated w ords give analogous results for the rational num b ers. F or example Theorems 2.1, 2.15 give the follo wing: Theorem 3.1. F or eve ry net { x q } q ∈ Q ∗ in a c omp act metric sp ac e ( X , d ) and ~ q = ( q n ) n ∈ N ∈ Q ∞ ( υ ) ther e exist an extr action ~ r = ( r n ) n ∈ N of ~ q an d x 0 ∈ X such that R 1 - lim q ∈ F S [ ( r n ( i n ,j n )) n ∈ N ] x q = x 0 (in p articular x r n ( i n ,j n ) → x 0 ), uniformly for every (( i n , j n )) n ∈ N ⊆ N × N with 1 ≤ i n , j n ≤ n . W e call a family { T q } q ∈ Q ∗ of con tin uous functions of a top ological space X in to it self a Q ∗ -system of X if T q 1 T q 2 = T q 1 + q 2 for q 1 < R 1 q 2 . Theorem 3.2. L et { T q 1 } q ∈ Q ∗ , . . . , { T q m } q ∈ Q ∗ b e m Q ∗ -systems of tr ansfo rm ations of a c omp act metric sp ac e X , al l c ontaine d in a c omm utative gr oup G of home omorphisms of X and let ~ q ∈ Q ∞ ( υ ) . Then, ther e exist x 0 ∈ X and an extr action ~ r = ( r n ) n ∈ N of ~ q such that, for every 1 ≤ i ≤ m , R 1 - lim q ∈ F S [ ( r n ( i n ,j n )) n ∈ N ] T q i ( x 0 ) = x 0 (in p articular, T r n ( i n ,j n ) i ( x 0 ) → x 0 ), uniformly for al l (( i n , j n )) n ∈ N ⊆ N × N with 1 ≤ i n , j n ≤ n . Mor e over, in c ase ( X, G ) is minimal, the set of such p oints x 0 is a d e n se subset of X. 16 Semigroup ( Z , +) . As w e describ ed in Example 1.1(2), for a giv en increasing sequence ( k n ) n ∈ N ⊆ N with k n ≥ 2 , the set Z ∗ of the nonzero integer num b ers can b e iden tified with a set L (Σ , ~ k ) of ω -lo cated words , via the function g : L (Σ , ~ k ) → Z ∗ , with g ( z s 1 . . . z s l ) = P l i =1 z s i ( − 1) s i − 1 l s i − 1 where l 0 = 1 and l s = k 1 . . . k s , for s > 0. W e extend the function g to the set L (Σ , ~ k ; υ ) of v ariable ω -lo cated words corresp onding to eac h w = z s 1 . . . z s l ∈ L (Σ , ~ k ; υ ) a function z = g ( w ) whic h sends ev ery i ∈ N with 1 ≤ i ≤ k min dom ( w ) , to z ( i ) = g ( T i ( w )) = P s ∈ C z s ( − 1) s − 1 l s − 1 + P s ∈ V i ( − 1) s − 1 l s − 1 . where C = { s ∈ d om ( w ) : z s ∈ Σ } and V = { s ∈ d om ( w ) : z s = υ } . Let Z ( υ ) = g ( L (Σ , ~ k ; υ )). Then the extended function g : L (Σ ∪ { υ } , ~ k ) → Z ∗ ∪ Z ( υ ) is one-to-one and onto. F or z 1 , z 2 ∈ Z ∗ ∪ Z ( υ ) w e define the relation z 1 < R 2 z 2 ⇐ ⇒ g − 1 ( z 1 ) < R 2 g − 1 ( z 2 ) . So, { x z } z ∈ Z ∗ ⊆ X , where X is a top ological space, can b e considered as an R 2 -net and consequen tly w e can define, for x 0 ∈ X , R 2 -lim z ∈ Z ∗ x z = x 0 iff for a n y neighbor- ho o d V of x 0 , there exists n 0 ≡ n 0 ( V ) ∈ N suc h tha t x z ∈ V for ev ery z ∈ Z ∗ with min dom ( g − 1 ( z )) ≥ n 0 . Observ e that g ( w 1 ⋆ w 2 ) = g ( w 1 ) + g ( w 2 ) for eve ry w 1 < R 2 w 2 ∈ L (Σ ∪ { υ } , ~ k ) . So , if ~ z = ( z n ) n ∈ N ∈ Z ∞ ( υ ) = { ( z n ) n ∈ N : z n ∈ Z ( υ ) and z n < R 2 z n +1 } , then the set of the extractions of ~ z is E V ∞ ( ~ z ) = { ~ v = ( v n ) n ∈ N ∈ Z ∞ ( υ ) : v n = g ( u n ) for ( u n ) n ∈ N ∈ E V ∞ (( g − 1 ( z n )) n ∈ N ) } and the set o f a ll the extracted inte gers of ~ z is E ( ~ z ) = { z ∈ F S [( z n ( i n )) n ∈ N ] : ( i n ) n ∈ N ⊆ N with 1 ≤ i n ≤ k n } = = { g ( w ) : w ∈ E (( g − 1 ( z n )) n ∈ N ) } . Of course, { x z } z ∈ E ( ~ z ) is an R 2 -subnet of { x z } z ∈ Z ∗ . Hence, via the function g , all the prese n ted results relating to ω - lo cated w ords give analogous results for t he integers . F or example Theorems 2.1 , 2.6 give the following. Theorem 3.3. F or every net { x z } z ∈ Z ∗ in a c o m p act metric sp a c e ( X , d ) , and ~ z = ( z n ) n ∈ N ∈ Z ∞ ( υ ) ther e exist an extr action ~ v = ( v n ) n ∈ N of ~ z and x 0 ∈ X such that R 2 - lim z ∈ F S [( v n ( i n )) n ∈ N ] x z = x 0 (in p articular x v n ( i n ) → x 0 ), uniformly for al l ( i n ) n ∈ N ⊆ N with 1 ≤ i n ≤ k n . 17 W e call a fa mily { T z } z ∈ Z ∗ of contin uous functions of a top ological space X into itself a Z ∗ -system of X if T z 1 T z 2 = T z 1 + z 2 for z 1 < R 2 z 2 . Theorem 3.4. L et { T z } z ∈ Z ∗ b e a Z ∗ -system of c ontinuous maps of a c omp act metric sp ac e ( X, d ) , ~ z = ( z n ) n ∈ N ∈ Z ∞ ( υ ) and y ∈ X . T h en ther e exist an extr action ~ v = ( v n ) n ∈ N of ~ z and x 0 ∈ X such that R 2 - lim z ∈ F S [( v n ( i n )) n ∈ N ] T z ( y ) = x 0 , R 2 - lim z ∈ F S [( v n ( i n )) n ∈ N ] T z ( x 0 ) = x 0 uniformly for al l ( i n ) n ∈ N ⊆ N with 1 ≤ i n ≤ k n . As w e described in Example 1 .1(3), the set of natural n um b ers can b e identifie d with a set L (Σ , ~ k ) and conseq uen tly all the presen ted results relating to ω -lo cat ed w ords give analogous recurrence results f o r t he natural num b ers. W e will no w giv e some applications of the previous ly men tioned recurrence results for systems or nets index ed b y w ords to systems or nets indexed by an arbitrar y semigroup. F or simplicit y w e will presen t only the case of comm utative semigroups. Let ( S, +) b e a semigroup and ( y l,n ) n ∈ Z ∗ ⊆ S f or ev ery l ∈ Z ∗ . Setting Σ = { α n : n ∈ Z ∗ } , where α n = n fo r n ∈ Z ∗ and ~ k = ( k n ) n ∈ Z ∗ ⊆ N , where ( k n ) n ∈ N and ( k − n ) n ∈ N are increasing seq uences, we define the function ϕ : e L (Σ , ~ k ) → S with ϕ ( w n 1 . . . w n m ) = P m i =1 y w n i ,n i . W e extend the function ϕ to the set e L (Σ , ~ k ; υ ) of v ariable w ords corresp onding to each w = w n 1 . . . w n m ∈ e L (Σ , ~ k ; υ ) a function s = ϕ ( w ) whic h sends ev ery ( i, j ) ∈ N × N with 1 ≤ j ≤ − max dom − ( w ) , 1 ≤ i ≤ min dom + ( w ) , to s ( i, j ) = ϕ ( T ( i,j ) ( w )) ∈ S . In case ( S, +) is a comm utativ e semigroup s ( i, j ) = ϕ ( w )(( i, j )) = P t ∈ C y w t ,t + P t ∈ V + y i,t + P t ∈ V − y − j,t , , where C = { n ∈ dom ( w ) : w n ∈ Σ } , V − = { n ∈ d om − ( w ) : w n = υ } and V + = { n ∈ dom + ( w ) : w n = υ } . F or a subset { x s : s ∈ S } of a top olo gical space X w e can consider the R 1 -net { x ϕ ( w ) } w ∈ e L (Σ , ~ k ) in X . Let ~ w = ( w n ) n ∈ N ∈ e L ∞ (Σ , ~ k ; υ ) suc h that R 1 -lim w ∈ e E ( ~ w ) x ϕ ( w ) = x 0 , for x 0 ∈ X . Then setting, for ev ery n ∈ N , s n = ϕ ( w n ) : { 1 , . . . , k n } × { 1 , . . . , k − n } → X with s n ( i, j ) = P t ∈ C n y w t ,t + P t ∈ V + n y i,t + P t ∈ V − n y − j,t , w e hav e that R 1 -lim s ∈ F S [( s n ( i n ,j n )) n ∈ N ] x s = x 0 18 uniformly for all (( i n , j n )) n ∈ N ⊆ N × N with 1 ≤ i n ≤ k n , 1 ≤ j n ≤ k − n . W e write R 1 -lim s ∈ F S [( s n ( i n ,j n )) n ∈ N ] x s = x 0 if and only if for an y neighborho o d V of x 0 , there exists n 0 ≡ n 0 ( V ) ∈ N suc h that x s ∈ V for ev ery s ∈ F S  s n ( i n , j n )  n ≥ n 0  . Hence, via the function ϕ , all the presen ted results related t o ω - Z ∗ -lo cated w ords giv e analogous results fo r nets indexed b y a n a r bit r a ry semigroup. F or example Theo- rems 2.1, 2.1 5 g iv e the following. Theorem 3.5. L et ( S, +) b e a c ommutative semig r oup and ( y l,n ) n ∈ Z ∗ ⊆ S for ev e ry l ∈ Z ∗ . F or eve ry subse t { x s : s ∈ S } o f a c omp act metric sp a c e ( X , d ) ther e exist x 0 ∈ X and, for every n ∈ N , functions s n : { 1 , . . . , k n } × { 1 , . . . , k − n } → X with s n ( i, j ) = P t ∈ C n y w t ,t + P t ∈ V + n y i,t + P t ∈ V − n y − j,t , wher e C n = C − n ∪ C + n ⊆ Z ∗ with max C − n +1 < min C − n < max C + n < min C + n +1 , V + n ⊆ N with max V + n < min V + n +1 and V − n ⊆ Z − with min V − n > max V − n +1 , such that R 1 - lim s ∈ F S [( s n ( i n ,j n )) n ∈ N ] x s = x 0 (in p articular x s n ( i n ,j n ) → x 0 ), uniformly for every (( i n , j n )) n ∈ N ⊆ N × N with 1 ≤ i n ≤ k n , 1 ≤ j n ≤ k − n . Corollary 3.6. L et ( S, +) b e a c ommutative semigr oup an d ( y n ) n ∈ Z ∗ ⊆ S . F or every subset { x s : s ∈ S } of a c omp act metric sp ac e ( X, d ) and functions p, q : N → N ther e e x ist x 0 ∈ X and ( a n ) n ∈ N ⊆ F S [( y n ) n ∈ Z ∗ ] , ( b n ) n ∈ N ⊆ F S [( y n ) n ∈ N ] and ( c n ) n ∈ N ⊆ F S [( y − n ) n ∈ N ] such that R 1 - lim s ∈ F S [( a n + p ( i n ) b n + q ( j n ) c n ) n ∈ N ] x s = x 0 (in p articular x a n + p ( i n ) b n + q ( j n ) c n → x 0 ) uniformly for every (( i n , j n )) n ∈ N ⊆ N × N with 1 ≤ i n , j n ≤ n . Pr o of. Set y l,n = p ( l ) y n for ev ery l ∈ N and y l,n = q ( − l ) y n for eve ry l ∈ Z − and apply Theorem 3.5.  Let ( S, +) b e a comm utativ e semigroup and ( y l,n ) n ∈ Z ∗ ⊆ S for ev ery l ∈ Z ∗ . W e call a family { T s } s ∈ S of con tinuous functions of a top ological space X into itself an e L (Σ , ~ k )- system of S if T ϕ ( w 1 ) T ϕ ( w 2 ) = T ϕ ( w 1 ⋆w 2 ) for w 1 < R 1 w 2 ∈ e L (Σ , ~ k ). Theorem 3.7. L et ( S, +) b e a c ommutative semigr oup, ( y l,n ) n ∈ Z ∗ ⊆ S fo r every l ∈ Z ∗ and { T s 1 } s ∈ S , . . . , { T s m } s ∈ S b e m e L (Σ , ~ k ) -systems of tr ans f o rmations of a c omp act metric sp ac e X , al l c ontaine d in a c ommutative gr oup G of home omorphisms of X . Then, ther e exist x 0 ∈ X and, fo r every n ∈ N , functions s n : { 1 , . . . , k n } × { 1 , . . . , k − n } → X with s n ( i, j ) = P t ∈ C n y w t ,t + P t ∈ V + n y i,t + P t ∈ V − n y − j,t , 19 wher e C n = C − n ∪ C + n ⊆ Z ∗ with max C − n +1 < min C − n < max C + n < min C + n +1 , V + n ⊆ N with max V + n < min V + n +1 and V − n ⊆ Z − with min V − n > max V − n +1 , such that R 1 - lim s ∈ F S [( s n ( i n ,j n )) n ∈ N ] T s i ( x 0 ) = x 0 for e v ery 1 ≤ i ≤ m, uniformly for every (( i n , j n )) n ∈ N ⊆ N × N with 1 ≤ i n ≤ k n , 1 ≤ j n ≤ n. Mor e over, in c ase ( X , G ) is minimal, the set of such p o ints x 0 is a dense subset of X . Ac kno wledgmen ts. W e thank Professor S. Negrep ontis for helpful discussions and supp ort during the preparation of this pap er. The first author a c know ledge par t ial sup- p ort from the Kap o distrias researc h grant of A thens Univ ersit y . The second author ac kno wledge partial suppo rt from the Stat e Sch olarship F oundation o f G reece. Reference s [Bi] M. V. Birkhoff, Dynamic al systems , Amer. Math. So c. Collo q. Publ. v ol. 9 (192 7). [BIP] T. Buda k, N. I¸ sik and J. Py m, Subsemigr oups of S tone- ˇ Ce ch c omp actific ations , Math. P ro c. Cambridge Phil. Soc. 11 6 (1994), 99–11 8 . [C] T. Carlson, Some unifying principles in R amsey the ory , Disc r ete Math. 68 (1988 ), 117–169. [F] V. F arma k i, R amsey The ory for Wor ds over an infinite A lphab et , arXiv: 0904 .1948 v1 [math.CO], (2009). [FK] V. F a r maki, A. Koutsogiannis, R amsey t he ory for wor ds r epr esent ing r ationals , arXiv: 013806 7 [math.CO], (2010). [F u] H. F urs ten b erg, R e curr enc e in Er go dic The ory and Combinatorial N umb er The ory , P rinceton Univ. Pres s , (1981). [F uKa] H. F urstenberg, Y. Katznelso n, Idemp otents in c omp act semigr oups and R amsey the ory , Israel J. Math. 68 , (198 9), 257–270 . [F uW ] H. F ur sten b erg, B. W eiss, T op olo gic al dynamics and c ombinatorial numb er the ory , J. d’Analyse Math. 34 , (197 8), 61–85. [H] N. Hindman, Finite sums fr om se quen c es within c el ls of a p artition of N , J. Co m binatorial Theory , Ser. A 17 (1974 ), 1–11 . [vdW] B. L. v an der W aerden, Beweis einer Baudetschen V ermutung , Nieuw Arch. Wisk . 15 (1927 ), 212–2 16. V a ssiliki F armaki: Dep ar tment of Ma thema tics, A thens University, P anepistemiopolis, 15784 A thens, Greece E-mail address: v fa rmaki@math.uoa.g r Andreas Koutsogia nnis: Dep ar tment of Ma thema tics, A thens University, P anepistemiopolis, 15784 A thens, Greece E-mail address: a k outsos@ math.uoa.gr 20