Rational Dynamical Systems

RA TIONAL D YNAMICAL SYSTEMS ANDREAS KOUTSOGIANNIS Abstract. W e int ro duce the notion of a rational dynamical system extending the clas- sical notion of a top ological dynamical system and we prov e (m ultiple) recurr e nc e results for such s ystems via a par titio n theorem for the r ational num b ers proved by F ar maki and the author in [FK]. In pa rticular, we extend classical recurrence results developed by F urstenber g and W eiss. Also, we give some applications of these topolo gical recurrence results to topolog y , to combinatorics, to diophan tine appro ximations and to num b er theory . Intr oduction In 1927, Birkhoff pro v ed (in [Bi]) that ev ery t o polo g ical d ynamical system ( X, T ) , where X is a compact metric space a nd T : X → X is a con tin uous map, ha s a recur- ren t elemen t, whic h means that there exist x ∈ X a nd a sequence of natural n umbers ( α n ) n ∈ N ⊆ N w ith α n → ∞ s uc h that T α n ( x ) → x. The m ultiple v ersion of Birkhoff ’s recurrence theorem is due to F ursten b erg and W eiss: Theorem 0.1. ([F uW], 1978 ) If X i s a c om p a ct m etric sp ac e an d T 1 , . . . , T l ar e c om - muting c ontinuous ma ps of X to itself, t hen ther e e x ists a p oint x ∈ X and a se quenc e ( α n ) n ∈ N ⊆ N , α n → ∞ such that T α n i ( x ) → x simultane ously for 1 ≤ i ≤ l (i n this c ase, x is sai d to b e a multiple r e curr e n t p oin t for T 1 , . . . , T l ). A consequence (in fact an equiv alen t form) o f Theorem 0.1 is the followin g: Theorem 0.2. L e t l ∈ N and ε > 0 . If X is a c omp act m etric sp ac e and T 1 , . . . , T l ar e c ommuting c ontinuous maps of X to itself, then ther e ex i s ts x 0 ∈ X and n 0 ∈ N , such that T n 0 i ( x 0 ) ∈ B ( x 0 , ε ) for every 1 ≤ i ≤ l . In fact, Theorem 0 .2 can b e considered as the top ological dynamics ve rsion of G allai’s partition theorem ( see [GRS]), namely that for l ∈ N and any finite partition o f N l , one of the cells of the pa rtition con tains an affine ima g e o f any finite subset of N l (an affine image of a finite su bset F of N l is any set of the form α + β F where α ∈ N l , β ∈ N ). 1991 Mathematics Subje ct Classific ation. Pr imary 37Bx x; 54H20. Key wor ds and phr ases. top ological dyna mics, recur rence, partition theorems, rational num b ers. 1 W e note that Gallai’s theorem is the m ultidimensional ex tension of v an de r W aerden’s theorem ([vdW], 1927) , that for any finite partition of the set o f natural n um b ers there exists a cell of the partition whic h contains arbitrary long arithmetic prog ress ions, whic h is a (p erhaps the most) fundamen tal result in R a mse y theory . Our starting p oin t for this pap er is a partitio n theorem for the set of rational n um b ers (Theorem 1.1 below ) pro v ed in [FK], whic h can b e c haracterized as a strong v an der W aerden theorem fo r t he set of rational n um b ers. This theorem follo ws from a more general partition theorem for w ords (in [FK]), using the fact that a rational num b er can b e r epresen ted as a w ord, as, according to a result of Budak-I¸ sik-Pym (in [BIP]), a rational n um b er q has a unique expression in the form 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 f o r ev ery r > 0 and q − s = q r = 0 f o r all but finite many r , s. Extending the classical notion of the top ological dynamical sys tem w e introduce the notion of a rationa l dy namical system (Definition 2.2). Consequen tly w e dev elop a r e- currence t heory for rational dynamic al systems, extending the fundamen tal results of F ursten b erg a nd W eiss fo r dynamical systems indexed by natural n um bers ([F u], [F uW]) stated ab o v e. W e remark that the presen ted recurrence results for rational systems a re stronger than those that follo ws from the more general recu rrence results concerning top ological dynamical systems indexed b y w ords presen ted in [FK2]. Specifically: (1) W e obtain a top ological v a n der W aerden theorem fo r the set of rational num b ers (Theorem 1.4) and its m ultiple v ersion (Theorem 3.1) extending Theorem 0.2 to rational dynamical systems . (2) Introducing the minimal rational systems and c haracterizing them as systems ha v- ing only uniformly rational recurren t p oin ts w e pro ve , in Theorem 3.2, a strong rec ur- rence pro perty o f minimal rational dynamical systems giving a n equiv alen t reformulation of Theorem 3.1. (3) W e obtain a strengthening of Theorem 0.1 f or ra tional dynamical systems in The- orem 4.1 and we prov e that it is an equiv alen t expression of Theorems 3.1 and 3.2. Also, w e presen t some applications of the previously men tioned results to topolo g y (Theorems 5.1 and 5.2), to com binatorics (Theorem 1.5, Theorem 3.3, Corollary 3.6), to diophantine approximations and to n um b er theory (applications of Theorems 5.1 2 and 5.2). F or example, as an application of Theorem 3.1 w e get a strong Gallai- t yp e partition theorem for the set of rational n um b ers (Theorem 3.3). W e will use the follow ing notation. Notation. Let N = { 1 , 2 , . . . } b e t he set of natura l n um b ers, Z = { . . . , − 2 , − 1 , 0 , 1 , 2 , . . . } the set o f in teger num b ers, Q = { m n : m ∈ Z , n ∈ N } the set o f rational n um b ers and Z − = { − n : n ∈ N } , Z ∗ = Z \ { 0 } , Q ∗ = Q \ { 0 } . W e denote b y [ X ] <ω > 0 the set of all the non-empt y finite subsets of X . F or a sequence ( x n ) n ∈ N of real n um bers we set F S  ( x n ) n ∈ N  = { P n ∈ F x n : F ∈ [ N ] <ω > 0 } . 1. A Topological v an der W aerden-type theore m f or the se t of ra tional numbers W e introduce the notion of a simple r a tional dynamical system defined from a compact metric space X and a sequence { T n } n ∈ Z ∗ of con tin uous functions from X to itself (Defi- nition 1.2). W e prov e a recurrence theorem for suc h systems in Theorem 1.4, extending the analogous result of F ursten b erg and W eiss (Theorem 0.2 , case l = 1 ) . Theorem 1.4 is an implication of a strengthened v an der W aerden-type theorem for the set of rational n um b ers prov ed in [FK] (see Theorem 1.1 below ). The inv erse implication is partially correct since Theorem 1.4 implies a we ak er form of Theorem 1.1, whic h actually can b e considered as a v an der W aerden-t yp e theorem for t he set of rational n um b ers. According to [BIP] ev ery ratio na l n um b er q has a unique expression in the form 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 f o r ev ery r > 0 and q − s = q r = 0 for all but finite man y r, s . So, f or ev ery non-zero ratio nal nu m b er q , there exists a unique l ∈ N , { t 1 < . . . < t l } = dom ( q ) ∈ [ Z ∗ ] <ω > 0 and { q t 1 , . . . , q t l } ⊆ N with 1 ≤ q t i ≤ − t i if t i < 0 and 1 ≤ q t i ≤ t i if t i > 0 for ev ery 1 ≤ i ≤ l , suc h that defining dom − ( q ) = { t ∈ do m ( q ) : t < 0 } and d om + ( q ) = { t ∈ dom ( q ) : t > 0 } to ha v e q = X t ∈ dom − ( q ) q t ( − 1) − t ( − t + 1)! + X t ∈ dom + ( q ) q t ( − 1) t +1 t ! (w e set X t ∈∅ = 0) . Observ e that e − 1 − 1 = − ∞ X t =1 2 t − 1 (2 t )! < X t ∈ dom − ( q ) q t ( − 1) − t ( − t + 1)! < ∞ X t =1 2 t (2 t + 1 )! = e − 1 3 and X t ∈ dom + ( q ) q t ( − 1) t +1 t ! ∈ Z ∗ if dom + ( q ) 6 = ∅ . F or t w o non-zero rational num b ers q 1 , q 2 w e set q 1 ≺ q 2 ⇐ ⇒ max dom − ( q 2 ) < min dom − ( q 1 ) < max d om + ( q 1 ) < min d om + ( q 2 ) . Using the previous represen tation of rational n um b ers we hav e the fo llo wing pa r t ition theorem for the rational n um b ers. Theorem 1.1. ( [ F K]) L et Q = Q 1 ∪ . . . ∪ Q r for r ∈ N . Then, t her e exist 1 ≤ i 0 ≤ r and fo r eve ry n ∈ N a function q n : { 1 , . . . , n } × { 1 , . . . , n } ∪ { (0 , 0) } → Q with q n ( i, j ) = X t ∈ C − n q n t ( − 1) − t ( − t + 1)! + i X t ∈ V − n ( − 1) − t ( − t + 1)! + X t ∈ C + n q n t ( − 1) t +1 t ! + j X t ∈ V + n ( − 1) t +1 t ! , wher e C − n , V − n ∈ [ Z − ] <ω > 0 , C + n , V + n ∈ [ N ] <ω > 0 with C − n ∩ V − n = ∅ = C + n ∩ V + n , q n t ∈ N with 1 ≤ q n t ≤ − t for t ∈ C − n , 1 ≤ q n t ≤ t for t ∈ C + n , wh ich satisfy q n ( i n , j n ) ≺ q n +1 ( i n +1 , j n +1 ) for every n ∈ N , and F S  q n ( i n , j n )  n ∈ N  ⊆ Q i 0 for al l (( i n , j n )) n ∈ N ⊆ N × N ∪ { (0 , 0) } with 0 ≤ i n , j n ≤ n for every n ∈ N . Definition 1.2. Let X b e a compact metric space and { T n } n ∈ Z ∗ a family of comm uting con tin uous maps of X to itself. F or ev ery non- zero rational n um b er q with dom ( q ) = { t 1 < . . . < t l } , we define T q ( x ) = T q t 1 t 1 . . . T q t l t l ( x ) and T 0 ( x ) = x for ev ery x ∈ X . W e sa y that T = ( T q ) q ∈ Q is a r ational ind exe d family define d f r om the c ontinuous maps { T n } n ∈ Z ∗ of X and ( X , T ) is a simple r ational dynamic al system . Remark 1.3. F or a rational indexed family T = ( T q ) q ∈ Q w e ha ve in general T p + q 6 = T p T q but if p, q ∈ Q ∗ with q ≺ p, then T p + q = T p T q . Using Theorem 1.1 we can prov e a recurrence result for simple rational dynamical systems e xtending the analogous res ult of F ursten b erg and W e iss (Theorem 0.2, case l = 1). Theorem 1.4. L et ( X, T ) b e a simp le r a tional dynamic al system, k ∈ N and k 1 < k 2 b e arbitr ary r e al numb ers. Th en, for every ε > 0 ther e exist β ∈ Q \ Z , γ ∈ Z ∗ with dom + ( β ) = ∅ = dom − ( γ ) , max dom − ( β ) < k 1 < k 2 < min dom + ( γ ) and x 0 ∈ X such that T pβ + q γ ( x 0 ) ∈ B ( x 0 , ε ) for every 0 ≤ p, q ≤ k . 4 Pr o of . Let k ∈ N and ε > 0 . Since X is compact, w e hav e X = S m i =1 B ( x i , ε 2 ) fo r some x 1 , . . . , x m ∈ X , m ∈ N . Let x ∈ X . W e form a partit io n of the set of rational n um b ers Q = Q 1 ∪ . . . ∪ Q m , where q ∈ Q i ⇔ T q ( x ) ∈ B ( x i , ε 2 ) a nd T q ( x ) / ∈ B ( x j , ε 2 ) fo r j < i. According to The orem 1.1, there exist 1 ≤ i 0 ≤ m and C − , V − , C + , V + non-empt y sets with C − ∩ V − = ∅ = C + ∩ V + , suc h that q ∗ ( p,q ) = X t ∈ C − q t ( − 1) − t ( − t + 1)! + p X t ∈ V − ( − 1) − t ( − t + 1)! + X t ∈ C + q t ( − 1) t +1 t ! + q X t ∈ V + ( − 1) t +1 t ! ∈ Q i 0 for ev ery 1 ≤ p, q ≤ k + 1 ≤ min { | t | : t ∈ V − ∪ V + } , where 1 ≤ q t ≤ | t | , t ∈ Z ∗ and max dom − ( q ∗ (1 , 1) ) < k 1 < k 2 < min dom + ( q ∗ (1 , 1) ) . Equiv alently , if β = P t ∈ V − ( − 1) − t ( − t +1)! , γ = P t ∈ V + ( − 1) t +1 t ! , δ = P t ∈ C − q t ( − 1) − t ( − t +1)! and ǫ = P t ∈ C + q t ( − 1) t +1 t ! w e hav e that T δ + p β + ǫ + q γ ( x ) ∈ B ( x i 0 , ε 2 ) fo r ev ery 1 ≤ p, q ≤ k + 1 . Let x 0 = T δ + β + ǫ + γ ( x ) . Then T pβ + q γ ( x 0 ) ∈ B ( x 0 , ε ) for ev ery 0 ≤ p, q ≤ k .  According to the previous pro of, Theorem 1.4 is an implication of Theorem 1.1. W e will sho w that the inv erse implication is par t ia lly corr ect. In fact w e will p oint out that Theorem 1.4 implies a w eak er fo r m of Theorem 1.1, whic h actually can b e considered as a v an der W aerden-ty p e theorem for the set of rational num b ers. So, Theorem 1 .4 can b e considered a s a t o polo gical v an der W aerden theorem for the set of ra tional num b ers. Theorem 1.5. L et k 1 < k 2 b e r e al numb ers. If Q = Q 1 ∪ . . . ∪ Q r , r ∈ N , then ther e exists 1 ≤ i 0 ≤ r s uch that the set Q i 0 has the fol lowin g pr op erty: for e very k ∈ N , ther e exist α ∈ Q , β ∈ Q \ Z an d γ ∈ Z ∗ with dom + ( β ) = ∅ = dom − ( γ ) , max do m − ( β ) < k 1 < k 2 < min dom + ( γ ) such that α + pβ + q γ ⊆ Q i 0 for every 0 ≤ p, q ≤ k . Pr o of . It is sufficien t to sho w that for ev ery k ∈ N some Q j satisfies the conclusion, for some Q i 0 will do fo r infinite k a nd that set will do for all k . Let k ∈ N , Ω = { 1 , . . . , r } Q and en umerate Q = S n ∈ N { q n } with q 1 = 0 . Ω b ecomes compact metric space with metric d ( ω , ω ′ ) = inf { 1 t : ω ( q i ) = ω ′ ( q i ) fo r 1 ≤ i < t } . Let T be a ra tional indexed family whic h is defined from { T n } n ∈ Z ∗ , where T n ω ( q ) = ω ( q + ( − 1) n +1 n !) , n ∈ N and T n ω ( q ) = ω ( q + ( − 1) − n ( − n +1)! ) , n ∈ Z − . 5 Define a specific p oin t ω ∈ Ω according to the rule ω ( q ) = i ⇔ q ∈ Q i and q / ∈ Q j for j < i and let X = {T s 1 . . . T s m ω : s i ∈ Q , m ∈ N , 1 ≤ i ≤ m } . Then, T is a r a tional indexed family of X . According to Theorem 1 .4 (for ε = 1) there exist β ∈ Q \ Z , γ ∈ Z ∗ with d om + ( β ) = ∅ = dom − ( γ ) , max dom − ( β ) < k 1 < k 2 < min dom + ( γ ) and x 0 ∈ X suc h that d ( T pβ + q γ ( x 0 ) , x 0 ) < 1 for ev ery 0 ≤ p, q ≤ k . Then, w e ha v e ( ∗ ) x 0 (0) = x 0 ( pβ + q γ ) for ev ery 0 ≤ p, q ≤ k . x 0 ∈ X , so, there exist s 1 , . . . , s m ∈ Q suc h that x 0 is close to T s 1 . . . T s m ω . Set α = s 1 + . . . + s m ∈ Q . According to ( ∗ ) w e ha ve that ω ( α ) = ω ( α + pβ + q γ ) for eve ry 0 ≤ p, q ≤ k , th us, we hav e α + pβ + q γ ∈ Q ω ( α ) for ev ery 0 ≤ p, q ≤ k .  Remark 1.6. W e note that Theorem 1.5 is not strong enough to prov e Theorem 1.4, mainly since w e cannot ha v e control to the supp ort of α ∈ Q relating to the supp ort of β ∈ Q \ Z , γ ∈ Z ∗ and since in g eneral T p + q 6 = T p T q . 2. Ra tional D ynamical Syste ms and Uniform Ra tional Recurrence In the presen t paragraph we intro duce the not ion o f a ratio nal dynamical system (Definition 2.2 b elo w) and also the notion of a uniform rat io nal recurren t p oin t of a suc h system (Definition 2.9). Defining the minimal rational dynamical systems (D efinition 2.4) w e prov e that ev ery rational dynamical system has uniform ra t io nal recurren t p oints . In fact a ratio nal dynamical system is minimal if a nd only if ev ery p oin t of t he system is uniform ra tional r ecurrent ( Theorems 2 .1 0 and 2.12). In order t o define the rationa l dynamical systems w e need the definition of the com- m uting rational indexed families whic h we give b elo w. Definition 2.1. Let T 1 = ( T q 1 ) q ∈ Q , . . . , T l = ( T q l ) q ∈ Q b e l rational indexed families of X defined from the comm ut ing families of maps { T 1 ,n } n ∈ Z ∗ , . . . , { T l,n } n ∈ Z ∗ of X resp ectiv ely . W e say that t he families T 1 , . . . , T l are c ommuting if T i,n 1 T j,n 2 = T j,n 2 T i,n 1 for ev ery 1 ≤ i, j ≤ l , n 1 , n 2 ∈ Z ∗ . Definition 2.2. Let X b e a compact metric space and { T 1 ,n } n ∈ Z ∗ , . . . , { T l,n } n ∈ Z ∗ b e l families o f commuting homeomorphisms from X to itself. If the indexed families T 1 = ( T q 1 ) q ∈ Q , . . . , T l = ( T q l ) q ∈ Q , whic h are defined from { T 1 ,n } n ∈ Z ∗ , . . . , { T l,n } n ∈ Z ∗ resp ectiv ely are commuting, then w e sa y that ( X , T 1 , . . . , T l ) is a r ational dynamic al system . Remark 2.3. F o r a rational indexed family T = ( T q ) q ∈ Q of a ratio nal dynamical system ( X , T ) we hav e in general tha t T − q 6 = ( T − 1 ) q , where we hav e set ( T − 1 ) q = ( T q ) − 1 . 6 W e will define and c haracterize the minimal r a tional dynamical systems. Definition 2.4. A rational dynamical system ( X , T 1 , . . . , T l ) , where T i = ( T q i ) q ∈ Q is defined fro m the comm uting homeomorphisms { T i,n } n ∈ Z ∗ of X for 1 ≤ i ≤ l , is said to b e minimal if for ev ery closed Y ⊆ X with T i,n ( Y ) = Y for ev ery 1 ≤ i ≤ l , n ∈ Z ∗ w e ha v e that Y = X or Y = ∅ . Prop osition 2.5. L et ( X, T 1 , . . . , T l ) b e a r a tion a l dynamic al system, wher e for 1 ≤ i ≤ l , T i = ( T q i ) q ∈ Q is define d fr om the c ommuting ho m e o morphisms { T i,n } n ∈ Z ∗ . L et G b e the gr oup of home omorphisms of X gener ate d by the functions T i,n , for n ∈ Z ∗ and 1 ≤ i ≤ l . The fo l lowing ar e e quivalent: (i) ( X, T 1 , . . . , T l ) is minim al. (ii) { S ( x ) : S ∈ G } = X for every x ∈ X. (iii) F or every non-em pty op en set V ⊆ X ther e exis t a non-empty finite subse t of G, F , such that S S ∈ F S − 1 ( V ) = X . Pr o of . ( i ) ⇒ ( ii ) Let x ∈ X. F or ev ery n ∈ Z ∗ , 1 ≤ i ≤ l, we hav e that T i,n ( { S ( x ) : S ∈ G } ) = { S ( x ) : S ∈ G } . Since { S ( x ) : S ∈ G } 6 = ∅ and ( X , T 1 , . . . , T l ) is a minimal dynamical system, w e ha v e that { S ( x ) : S ∈ G } = X . ( ii ) ⇒ ( i ) If Y is a closed non- empt y closed subset of X with T i,n ( Y ) = Y for ev ery n ∈ Z ∗ , 1 ≤ i ≤ l, then X = { S ( y ) : S ∈ G } ⊆ Y for ev ery y ∈ Y . Then Y = X , th us, ( X , T 1 , . . . , T l ) is minimal. ( i ) ⇒ ( iii ) F or eve ry non- empt y o pen set V w e hav e S S ∈ G S − 1 ( V ) = X. F rom the compactness o f X w e hav e the conclusion. ( iii ) ⇒ ( i ) Let ( X , T 1 , . . . , T l ) is not minimal. Let Y b e a non-empt y closed in v ariant prop er subset of X and V = X \ Y . Then S S ∈ G S − 1 ( V ) 6 = X , a con tradiction.  Definition 2.6. Let ( X , T 1 , . . . , T s ) b e a rational dynamical system and Y ⊆ X . W e say that the system ( Y , T 1 | Y , . . . , T s | Y ) is a subsystem of ( X , T 1 , . . . , T s ) if ( i ) Y is a closed subset of X , and ( ii ) T i,n ( Y ) = Y for ev ery 1 ≤ i ≤ s, n ∈ Z ∗ . Prop osition 2.7. Every r ation a l dynamic al system has a mi n imal subsystem. Pr o of . Let ( X, T 1 , . . . , T l ) b e a rational dynamical syste m, where for 1 ≤ i ≤ l, T i = ( T q i ) q ∈ Q is defined f rom the comm uting homeomorphisms { T i,n } n ∈ Z ∗ . Let L = { Y ⊆ X : Y 6 = ∅ , Y closed and T i,n ( Y ) = Y fo r ev ery n ∈ Z ∗ , 1 ≤ i ≤ l } . L 6 = ∅ since X ∈ L . Let 7 D ⊆ L b e a family to taly ordered by inclusion. D has the finite interse ction prop ert y and since X is compact w e hav e that A := T Y ∈D Y 6 = ∅ , with A ⊆ Y for ev ery Y ∈ D . According t o Zorn’s lemma there exists a minimal Y 0 ∈ L . Then, ( Y 0 , T 1 | Y 0 , . . . , T l | Y 0 ) is a minimal subsystem.  W e will intro duce the notion of uniformly rational recurren t p oin ts for a ratio nal dy- namical system. Firstly w e will rem ind the notion of a syndetic subset of an a belian (semi-)group. Definition 2.8. A subset E of an ab elian (semi-)group G is synde tic if there exists F ∈ [ G ] <ω > 0 suc h that G = S g ∈ F { s ∈ G : g + s ∈ E } . Definition 2.9. Let ( X , T 1 , . . . , T l ) b e a rational dynamical system, where for 1 ≤ i ≤ l , T i is defined from the comm uting homeomorphisms { T i,n } n ∈ Z ∗ . A p oin t x ∈ X is uniformly r ational r e curr ent for ( X, T 1 , . . . , T l ) if for a ny neigh b orho o d V of x, the se t { S ∈ G : S ( x ) ∈ V } is syndetic, where G is the group of homeomorphisms o f X generated b y the functions T i,n for ev ery n ∈ Z ∗ and 1 ≤ i ≤ l . The following theorem giv es the connection b et wee n minimal rational dynamical sys- tems a nd uniformly ra tional recurren t p oints . Theorem 2.10. I f ( X, T 1 , . . . , T l ) is a minimal r ational dynam ic al system, then every p oint x ∈ X is uniform l y r ational r e curr ent. Pr o of . If T i , for 1 ≤ i ≤ l , is defined from t he commuting homeomorphisms { T i,n } n ∈ Z ∗ and G is the group of homeomorphisms of X generated by the functions T i,n for ev ery n ∈ Z ∗ and 1 ≤ i ≤ l , then for ev ery x ∈ X and ev ery non-empty open set V ⊆ X , the set { S ∈ G : S ( x ) ∈ V } is syndetic. Indeed, according to Prop osition 2 .5 w e hav e that S m i =1 S − 1 i ( V ) = X for some m ∈ N , S 1 , . . . , S m ∈ G. So, for ev ery S ∈ G w e hav e S i ( S ( x )) ∈ V for some 1 ≤ i ≤ m, or S i S ∈ { S ∈ G : S ( x ) ∈ V } .  Corollary 2.11. F or any r ational dynamic al system ( X , T 1 , . . . , T l ) , the set of uniformly r ational r e curr ent p oints is non-e m pty. Pr o of . Immediate fr om Prop osition 2.7 and Theorem 2.10.  No w w e can c haracterize the minimal subsystem s of a ratio nal dynamical system via the uniformly rational recurren t p oin ts of the system. 8 Theorem 2.12. L et ( X, T 1 , . . . , T l ) b e a r ational dynamic al system, wher e for 1 ≤ i ≤ l , T i is defi n e d fr om the c ommuting home omorphisms { T i,n } n ∈ Z ∗ and G b e the gr oup of home om orphisms of X gener ate d by the functions T i,n for every n ∈ Z ∗ and 1 ≤ i ≤ l . Then ( Y , T 1 | Y , . . . , T l | Y ) is a minim al subsystem of ( X , T 1 , . . . , T l ) if an d only if Y = { S ( x ) : S ∈ G } for x a uniform l y r ational r e curr ent p oint of ( X , T 1 , . . . , T l ) . Pr o of . It suffices to prov e that if y ∈ { S ( x ) : S ∈ G } then x b elongs to { S ( y ) : S ∈ G } . Assume otherwise and let V b e an op en neigh b o rhoo d of x with V ∩ { S ( y ) : S ∈ G } = ∅ . x is a uniformly rational r ecurrent p oin t, so there exists a finite set { S 1 , . . . , S m } , m ∈ N of elemen ts of G suc h that for ev ery S ∈ G to ha v e S i ( S ( x )) ∈ V for some 1 ≤ i ≤ m. So, { S ( x ) : S ∈ G } ⊆ S m i =1 S − 1 i ( V ) , thus y ∈ { S ( x ) : S ∈ G } ⊆ S m i =1 S − 1 i ( V ) . Then w e ha v e V ∩ { S ( y ) : S ∈ G } 6 = ∅ , a con tradiction.  3. The recurrence proper ties of ra tional dynamical systems In Theorems 3.1 and 3.2 b elo w, w e prov e that rationa l dynamical systems has signif- ican t recurrence prop erties, analogous to those of the clas sical dynamical sys tems. So, Theorem 3.1 is an extension of Theorem 0.2 of F urstenberg-W eiss to rational dynam- ical s ystems and Theorem 3.2 is an equiv alen t reformulation of The orem 3.1 (fo r the analogous results for systems indexed b y N or Z see [F u], [F uW] and [M]). As a consequence of Theorem 3.1, whic h is a mu ltiple v ersion of Theorem 1.4 in case the transformatio ns T i are in v ertible, w e get a G allai-t yp e com binatorial result for the rational n um b ers (Theorem 3.3), pro ving tha t for l ∈ N and any finite partition of Q l , one of the cells of the pa rtition con tains (generalized) affine images o f ev ery finite subset of Q l . W e also remark that syndetic subsets of Q l ha v e the same prop ert y and that Theorem 3.3 has implications for functions on large c h unks of Q l . Theorem 3.1. L e t l ∈ N and ( X, T 1 , . . . , T l ) a r a tional dynamic al system, k ∈ N a n d k 1 < k 2 b e arbitr ary r e al numb ers. F or every ε > 0 ther e exists β ∈ Q \ Z , γ ∈ Z ∗ with dom + ( β ) = ∅ = dom − ( γ ) , max dom − ( β ) < k 1 < k 2 < min dom + ( γ ) and x 0 ∈ X such that T pβ + q γ i ( x 0 ) ∈ B ( x 0 , ε ) for every 0 ≤ p, q ≤ k , 1 ≤ i ≤ l . Theorem 3.2. L et l ∈ N , ( X, T 1 , . . . , T l , R ) b e a mini mal r ational dynamic al system, k ∈ N and k 1 < k 2 b e arbitr ary r e al numb ers. Then for every no n-empty op en set U ther e exist β ∈ Q \ Z and γ ∈ Z ∗ with dom + ( β ) = ∅ = dom − ( γ ) , max dom − ( β ) < k 1 < k 2 < 9 min dom + ( γ ) such that \ 0 ≤ p,q ≤ k ( U ∩ ( T pβ + q γ 1 ) − 1 U ∩ . . . ∩ ( T pβ + q γ l ) − 1 U ) 6 = ∅ . Pr o of of The o r em s 3.1 and 3.2. Our metho d of pro of is induction on l and consists of three steps: (1) W e will sho w that Theorem 3.1 holds for l = 1 , (2) if Theorem 3.1 holds for some l ∈ N then Theorem 3 .2 also holds for l , a nd (3) if Theorem 3.2 holds for some l ∈ N then Theorem 3 .1 holds fo r l + 1 . F or l = 1 we ha v e the conclusion of Theorem 3.1 from Theorem 1 .4. Let l ∈ N suc h that Theorem 3.1 holds. Let ( X , T 1 , . . . , T l , R ) a minimal rational dynamical system, where T i is defined from the comm uting homeomorphisms { T i,n } n ∈ Z ∗ of X for 1 ≤ i ≤ l , R is de fined from the comm uting homeomorphisms { R n } n ∈ Z ∗ of X , k ∈ N and k 1 < k 2 b e arbitra r y real n um b ers. Let U ⊆ X a non-empt y op en set. There exists u ∈ U and ε > 0 suc h that B ( u , ε ) ⊆ U. Let V = B ( u, ε 2 ) ⊆ U. Then, for ev ery x ∈ X with d ( x, V ) := inf { d ( x, y ) : y ∈ V } < ε 2 w e ha v e that x ∈ U. Let G b e the group of ho meomorphisms generated b y { T i,n } n ∈ Z ∗ , 1 ≤ i ≤ l a nd { R n } n ∈ Z ∗ . Since the system ( X , T 1 , . . . , T l , R ) is minimal there exists some m ∈ N , S 1 , . . . , S m ∈ G with X = S m i =1 S − 1 i V ( ∗ ) . Since X is compact, ev ery S i , 1 ≤ i ≤ m is uniformly con tin- uous, so there exists δ > 0 suc h that if y , z ∈ X with d ( y , z ) < δ then d ( S i ( y ) , S i ( z )) < ε 2 for 1 ≤ i ≤ m. According to Theorem 3.1 there exis t y ∈ X , β ∈ Q \ Z and γ ∈ Z ∗ with dom + ( β ) = ∅ = dom − ( γ ) , max dom − ( β ) < k 1 < k 2 < min dom + ( γ ) suc h that d ( y , T pβ + q γ i ( y )) < δ for ev ery 0 ≤ p, q ≤ k , 1 ≤ i ≤ l. F rom ( ∗ ) we hav e that there exists 1 ≤ j ≤ m such that y ∈ S − 1 j V . Set x = S j ( y ) ∈ V . Since S j com- m utes with the { T i,n } n ∈ Z ∗ , for 1 ≤ i ≤ l, w e ha v e that d ( x, T pβ + q γ i ( x )) < ε 2 for ev ery 0 ≤ p, q ≤ k , 1 ≤ i ≤ l . Then we hav e that { x, T pβ + q γ 1 ( x ) , . . . , T pβ + q γ l ( x ) } ⊆ U for ev ery 0 ≤ p, q ≤ k , so x ∈ T 0 ≤ p,q ≤ k ( U ∩ ( T pβ + q γ 1 ) − 1 U ∩ . . . ∩ ( T pβ + q γ l ) − 1 U ) 6 = ∅ , the conclusion. Let that Theorem 3.2 holds for some l ∈ N . W e will sho w that Theorem 3.1 holds for l + 1 . Let ( X , T 1 , . . . , T l +1 ) a rational dynamical system, where T i is defined fro m the comm uting homeomorphisms { T i,n } n ∈ Z ∗ of X for 1 ≤ i ≤ l + 1 , k ∈ N and k 1 < k 2 b e arbitrary real n um b ers. Witho ut loss of generality w e can supp ose that ( X , T 1 , . . . , T l +1 ) is minimal ( o r else w e replace ( X , T 1 , . . . , T l +1 ) with a minimal subs ystem). Let U 0 a non-empt y op en set with diam ( U 0 ) := sup { d ( x, y ) : x, y ∈ U 0 } < ε 2 . According to Theorem 3 .2 (f o r the minimal system ( X , T 1 T − 1 l +1 , . . . , T l T − 1 l +1 , T l +1 )) there exist β 1 ∈ Q \ Z , γ 1 ∈ Z ∗ with dom + ( β 1 ) = ∅ = d om − ( γ 1 ) , max dom − ( β 1 ) < k 1 < k 2 < min dom + ( γ 1 ) suc h 10 that B 0 := \ 0 ≤ p,q ≤ k ( U 0 ∩ l \ s =1 [ T pβ 1 + q γ 1 s ( T pβ 1 + q γ 1 l +1 ) − 1 ] − 1 U 0 ) 6 = ∅ . Let U 1 a no n- empt y op en set with diam ( U 1 ) < ε 2 suc h that U 1 ⊆ T 0 ≤ p,q ≤ k ( T pβ 1 + q γ 1 l +1 ) − 1 B 0 = T 0 ≤ p,q ≤ k T l +1 s =1 ( T pβ 1 + q γ 1 s ) − 1 U 0 . Supp ose that for m ∈ N w e ha v e c hosen U 1 , . . . , U m non-empt y , op en sets with d iam ( U i ) < ε 2 for ev ery 1 ≤ i ≤ m, suc h that ( ∗∗ ) U j ⊆ \ 0 ≤ p,q ≤ k l +1 \ s =1 ( T p ( β j + ... + β i +1 )+ q ( γ j + ... + γ i +1 ) s ) − 1 U i for ev ery 0 ≤ i < j ≤ m, with β j − 1 + γ j − 1 ≺ β j + γ j for 2 ≤ j ≤ m, dom + ( β j ) = ∅ = dom − ( γ j ) fo r 1 ≤ j ≤ m. F rom Theorem 3.2 there exist β m +1 ∈ Q \ Z and γ m +1 ∈ Z ∗ with β m + γ m ≺ β m +1 + γ m +1 , dom + ( β m +1 ) = ∅ = dom − ( γ m +1 ) such that B m := \ 0 ≤ p,q ≤ k ( U m ∩ l \ s =1 [ T pβ m +1 + q γ m +1 s ( T pβ m +1 + q γ m +1 l +1 ) − 1 ] − 1 U m ) 6 = ∅ . Let U m +1 a no n- empt y op en set with diam ( U m ) < ε 2 and U m +1 ⊆ T 0 ≤ p,q ≤ k ( T pβ m +1 + q γ m +1 l +1 ) − 1 B m = T 0 ≤ p,q ≤ k T l +1 s =1 ( T pβ m +1 + q γ m +1 s ) − 1 U m . Us ing this and ( ∗ ∗ ) for j = m w e hav e t ha t for ev ery 0 ≤ i ≤ m, U m +1 ⊆ \ 0 ≤ p,q ≤ k l +1 \ s =1 ( T p ( β m +1 + ... + β i +1 )+ q ( γ m +1 + ... + γ i +1 ) s ) − 1 U i . Inductiv ely w e can suppose that w e hav e sequence s ( U n ) n ∈ N ∪{ 0 } , ( β n ) n ∈ N and ( γ n ) n ∈ N with β n + γ n ≺ β n +1 + γ n +1 , d om + ( β n ) = ∅ = dom − ( γ n ) f or ev ery n ∈ N , suc h tha t ( ∗∗ ) holds for ev ery m ∈ N , with β j + . . . + β i +1 ∈ Q \ Z and γ j + . . . + γ i +1 ∈ Z ∗ , for ev ery 0 ≤ i < j ≤ m, m ∈ N . F or ev ery n ∈ N ∪ { 0 } let x n ∈ U n . Since X is seque n tial c ompact there exists i 0 < j 0 suc h that d ( x i 0 , x j 0 ) < ε 2 . According to ( ∗∗ ) , if we set β = β j 0 + . . . + β i 0 +1 ∈ Q \ Z and γ = γ j 0 + . . . + γ i 0 +1 ∈ Z ∗ , w e ha ve that { T pβ + q γ 1 ( x j 0 ) , . . . , T pβ + q γ l +1 ( x j 0 ) } ⊆ U i 0 for ev ery 0 ≤ p, q ≤ k . Also, x i 0 ∈ U i 0 , d ( x i 0 , x j 0 ) < ε 2 and diam ( U i 0 ) < ε 2 , th us for x = x j 0 , w e hav e that T pβ + q γ i ( x ) ∈ B ( x, ε ) for ev ery 0 ≤ p, q ≤ k , 1 ≤ i ≤ l + 1 . The pro of is complete.  W e ha v e already seen that Theorem 1.4 implies a v an der W aerden-ty p e theorem for the set of ra tional n umbers (Theorem 1.5). Ga lla i extended v an der W aerden theorem to higher dimensions and later F ursten b erg a nd W eiss g a ve a pro of of this result ([F uW]), 11 using to polo gical dynamics theory . Using Theorem 3 .1 w e will state and prov e, in The- orem 3.3, a Gallai- t yp e partitio n theorem for the set Q l , l ∈ N , using metho ds from [F u]. Note that Theorem 3.3 can b e considered a s a geometric ve rsion of Theorem 3.1 . Theorem 3.3. L et l ∈ N and k 1 < k 2 b e arbitr ary r e al numb ers. If Q l = Q 1 ∪ . . . ∪ Q r , r ∈ N , then ther e ex i s ts 1 ≤ i 0 ≤ r such that the set Q i 0 has the pr op erty that for k ∈ N , if F ∈ [ Q l ] <ω > 0 , then ther e exists α ∈ Q l , β ∈ Q \ Z and γ ∈ Z ∗ with d om + ( β ) = ∅ = dom − ( γ ) , max dom − ( β ) < k 1 < k 2 < min dom + ( γ ) such that α + ( pβ + q γ ) F ⊆ Q i 0 for every 0 ≤ p, q ≤ k . Pr o of . Let Q l = Q 1 ∪ . . . ∪ Q r , r ∈ N . It suffices to pr o duce t he set Q j for a given configuration F and k ∈ N . F or since there are o nly finite many p ossibilities for Q j and since a sequence F n ma y b e c hosen where eac h con tains all the preceding ones and any F is con tained in one of them, a set Q j that o ccurs for infinitely man y F n and k ∈ N will w ork for all F and all k . That would b e the desired Q i 0 . So we assume that a finite subset of Q l , F = { e e 1 , . . . , e e m } a nd k ∈ N are giv en. Let Ω = { 1 , . . . , r } Q l and enume rate Q = S n ∈ N { q n } with q 1 = 0 . Then Ω b ecomes a compact metric space if we define a metric b y: d ( ω , ω ′ ) = inf { 1 t : ω ( q i 1 , . . . , q i l ) = ω ′ ( q i 1 , . . . , q i l ) fo r 1 ≤ i 1 , . . . , i l < t } . F or 1 ≤ i ≤ m and e q ∈ Q l , let for n ∈ N , T i,n ω ( e q ) = ω ( e q + ( − 1) n +1 n ! e e i ) and for n ∈ Z − , T i,n ω ( e q ) = ω ( e q + ( − 1) − n ( − n +1)! e e i ) . F or 1 ≤ i ≤ m we fo rm the rational indexed family T i whic h is defined from the comm ut- ing ho meomorphism s { T i,n } n ∈ Z ∗ . W e define a sp ecific p oint ω ∈ Ω by ω ( e q ) = i ⇔ e q ∈ Q i and e q / ∈ Q j for j < i and let X = {T s 1 , 1 1 . . . T s 1 ,l 1 1 . . . T s m, 1 m . . . T s m,l m m ω , s i,j ∈ Q , l i ∈ N , 1 ≤ j ≤ l i , 1 ≤ i ≤ m } . Then, ( X , T 1 , . . . , T m ) is a ra tional dynamical system, so, according to Theorem 3.1 (for ε = 1) there exists β ∈ Q \ Z , γ ∈ Z ∗ with dom + ( β ) = ∅ = d om − ( γ ) , max dom − ( β ) < k 1 < k 2 < min dom + ( γ ) and x 0 ∈ X suc h that d ( T pβ + q γ i ( x 0 ) , x 0 ) < 1 f o r ev ery 0 ≤ p, q ≤ k , 1 ≤ i ≤ m. Th us ( ∗ ) x 0 ((0 , . . . , 0)) = x 0 (( pβ + q γ ) e e 1 ) = . . . = x 0 (( pβ + q γ ) e e m ) fo r ev ery 0 ≤ p, q ≤ k . x 0 ∈ X, so it is arbitrary close to some translate o f ω , T s 1 , 1 1 . . . T s 1 ,l 1 1 . . . T s m, 1 m . . . T s m,l m m ω , for some s i,j ∈ Q , l i ∈ N , 1 ≤ j ≤ l i , 1 ≤ i ≤ m. 12 Let e α = ( s 1 , 1 + . . . + s 1 ,l 1 ) e e 1 + . . . + ( s m, 1 + . . . + s m,l m ) e e m . It f o llo ws from ( ∗ ) tha t ω ( e α ) = ω ( e α + ( pβ + q γ ) e e 1 ) = . . . = ω ( e α + ( pβ + q γ ) e e m ) fo r ev ery 0 ≤ p, q ≤ k , so, we hav e e α + ( pβ + q γ ) F ⊆ Q ω ( e α ) for ev ery 0 ≤ p, q ≤ k .  Remark 3.4. Ac cording t o the pro of of the previous theorem, w e see that w e cannot ha v e con trol to the domain of the co ordinates of α ∈ Q l (as in the pro of o f Theorem 1.4 ) . F or this reason and also since for a rational indexed family T , w e ha v e in general that T p + q 6 = T p T q ( p, q ∈ Q ) efforts to pro v e Theorem 3.1 or 3.2 from Theorem 3.3 (proving sim ultaneously the equiv alence o f these theorems) w ere fruitless. Let giv e some corollaries of Theorem 3.3. Definition 3.5. Let l ∈ N and k 1 < k 2 b e arbitrary real n um b ers. W e sa y that the subset B ⊆ Q l is a R VDW(l,k 1 ,k 2 ) -set if for ev ery F ∈ [ Q l ] <ω > 0 and k ∈ N there exist α ∈ Q l , β ∈ Q \ Z and γ ∈ Z ∗ with dom + ( β ) = ∅ = dom − ( γ ) , max dom − ( β ) < k 1 < k 2 < min dom + ( γ ) suc h that α + ( pβ + q γ ) F ⊆ B for ev ery 0 ≤ p, q ≤ k . W e will no w prov e that syndetic sets b elongs to the previous family . Corollary 3.6. L e t l ∈ N and k 1 < k 2 b e arbitr ary r e al numb ers. If E is a syndetic subset of Q l , then E is a R VD W(l,k 1 ,k 2 )-set. Pr o of . Let E be a syndetic subset of Q l and k 1 < k 2 b e arbitrary real n um b ers. Then, Q l = S x ∈ F ( E + x ) f o r some F ∈ [ Q l ] <ω > 0 . According to Theorem 3.3 there exists x 0 ∈ F suc h that E + x 0 is a R VD W( l, k 1 , k 2 )-set. So, E is a R VD W( l , k 1 , k 2 )-set, since this prop ert y is tra nslatio n in v a rian t .  Theorem 3.3 has implications for functions o n large ch unks of Q l . Corollary 3.7. L et F ∈ [ Q l ] <ω > 0 , l ∈ N , k 1 < k 2 b e arbitr ary r e al numb ers, k ∈ N and r ∈ N . Ther e exis t n 0 ≡ n 0 ( l , k , k 1 , k 2 , r , F ) ∈ N such that, if Q l n , n ∈ N denotes the set of ve c tors in Q l with c omp onents b etwe en − n and n, then when ever n ≥ n 0 and Q l n = Q 1 ∪ . . . ∪ Q r , ther e exist 1 ≤ i 0 ≤ r , α ∈ Q l , β ∈ Q \ Z and γ ∈ Z ∗ with dom + ( β ) = ∅ = dom − ( γ ) , max dom − ( β ) < k 1 < k 2 < min dom + ( γ ) such that α + ( pβ + q γ ) F ⊆ Q i 0 for every 0 ≤ p, q ≤ k . Pr o of . Supp ose that for n → ∞ w e can find par t itions suc h tha t the conclusion do esn’t hold. Consider the corresp onding functions from Q l n to Λ = { 1 , . . . , r } whic h are defined 13 from these partitio ns and extend them arbitrary to Q l to obtain a p oin t ω n ∈ Λ Q l . Let ω a limit p oin t of ( ω n ) n ∈ N . According to Theorem 3.3 t here exist α ∈ Q l , β ∈ Q \ Z and γ ∈ Z ∗ with d om + ( β ) = ∅ = dom − ( γ ) , max dom − ( β ) < k 1 < k 2 < min dom + ( γ ) suc h that ω is constan t on α + ( pβ + q γ ) F for ev ery 0 ≤ p, q ≤ k . F or n large enough, w e ha v e that α + ( pβ + q γ ) F ⊆ Q l n for ev ery 0 ≤ p, q ≤ k and t ha t for n ′ large ω n ′ | Q l n = ω | Q l n , a con tradiction.  Notes. Since (as w e already no ticed in Remark 1.3 and R emark 3.4) for a rational indexed family T = ( T q ) q ∈ Q , w e hav e in general that T p + q 6 = T p T q ( p, q ∈ Q ), w e can’t ha v e (with these metho ds) p olynomial extensions of the results of this parag raph. 4. A Furstenbe r g-Weiss-typ e The orem for ra tional dynamical sys tems In this section we will pr ov e (in Theorem 4.1) a strengthening of Theorem 0.1 f o r ra- tional dynamical system s, namely , w e pro v e that if ( X, T 1 , . . . , T l ) is a rational dynamical system, k ∈ N and k 1 < k 2 are arbitrary real n umbers that there exist x ∈ X and se- quences ( β n ) n ∈ N ⊆ Q \ Z , ( γ n ) n ∈ N ⊆ Z ∗ with dom + ( β 1 ) = ∅ = dom − ( γ 1 ) , max dom − ( β 1 ) < k 1 < k 2 < min dom + ( γ 1 ) , β n + γ n ≺ β n +1 + γ n +1 , do m + ( β n ) = ∅ = dom − ( γ n ) fo r every n ∈ N suc h that T pβ n + q γ n i ( x ) → x for ev ery 0 ≤ p, q ≤ k simultaneous ly for 1 ≤ i ≤ l ( we call these p oints multiple r ational r e curr ent p oints ). Moreov er w e pro v e that Theorem 4.1 is equiv alen t to Theorems 3.1 and 3.2 and also tha t the m ultiple rational recurren t p oin ts consist a residual subset of X (Definition 4.2). A t this p oin t (as in [F u] for the analogous dynamical systems related to N or Z ) observ e that if the condition of comm utativit y of the system is o mitt ed, the conclusion do es not holt. F or example, let X = R ∪ {∞} , T n ( x ) = x 2 , n ∈ Z ∗ and S n ( x ) = x + 1 , n ∈ Z ∗ . If T and S are defined from { T n } n ∈ Z ∗ and { S n } n ∈ Z ∗ resp ectiv ely , then, the only recurren t p oin t o f T is 0 and the o nly one for S is ∞ . Also, without comm utativit y it may still happ en that the return times o f an y p oint are disjoin t for the v a rious transformations. F or example, let X = {− 1 , 1 } Q and T b e the indexed family which is defined from { T n } n ∈ Z ∗ with T n ω ( q ) = ω ( q + ( − 1) n +1 n !) , n ∈ N and T n ω ( q ) = ω ( q + ( − 1) − n ( − n +1)! ) , n ∈ Z − . Let R : X → X with R ( ω ( q ) ) = ω ( q ) if q = 0 and R ( ω ( q )) = − ω ( q ) if q 6 = 0 and let S n = RT n R, n ∈ Z ∗ . Then, if S is the indexed family whic h is defined from { S n } n ∈ Z ∗ , w e hav e tha t S q = R T q R for ev ery q ∈ Q . T q ω close to ω implies that ω ( q ) = ω (0) . S q ω 14 close to ω implies that T q Rω is close to R ω , so Rω ( q ) = R ω (0) , th us − ω ( q ) = ω ( 0) if q 6 = 0 . W e ha v e that T q ω and S q ω cannot b e sim ultaneously close to ω . A strengthening of Theorem 0.1 related to rational nu m b ers is the following. Theorem 4.1. L et ( X, T 1 , . . . , T l ) b e a r ational dynamic al system, k ∈ N a n d k 1 < k 2 b e arbitr ary r e al numb ers. Ther e exis ts a x ∈ X and se quenc es ( β n ) n ∈ N ⊆ Q \ Z , ( γ n ) n ∈ N ⊆ Z ∗ with dom + ( β 1 ) = ∅ = d om − ( γ 1 ) , max dom − ( β 1 ) < k 1 < k 2 < min dom + ( γ 1 ) , β n + γ n ≺ β n +1 + γ n +1 , dom + ( β n ) = ∅ = dom − ( γ n ) for every n ∈ N such that T pβ n + q γ n i ( x ) → x for every 0 ≤ p, q ≤ k sim ultane ously fo r 1 ≤ i ≤ l . Pr o of . F o r ev ery s > 0 let F s = { x ∈ X : there exists β ∈ Q \ Z , γ ∈ Z ∗ with dom + ( β ) = ∅ = dom − ( γ ) , max dom − ( β ) < k 1 < k 2 < min dom + ( γ ) suc h that d ( T pβ + q γ i ( x ) , x ) < 1 s for ev ery 0 ≤ p, q ≤ k , 1 ≤ i ≤ l } . If the conclusion is not true, t hen X = S n ∈ N ( X \ F n ) . W e claim that for ev ery n ∈ N w e ha v e that ( X \ F n ) ◦ = ∅ , a con tradiction according to Baire’s Category Theorem, since ev ery X \ F n is closed. Supp ose that ( X \ F n 0 ) ◦ 6 = ∅ for some n 0 ∈ N . Without loss of generality we can supp ose that ( X , T 1 , . . . , T l ) is minimal. F or 1 ≤ i ≤ l , let T i b e defined from the comm uting ho meomorphisms { T i,n } n ∈ Z ∗ of X . If G is the g roup of ho meomorphisms generated b y { T i,n } n ∈ Z ∗ , 1 ≤ i ≤ l , then X = S − 1 1 ( X \ F n 0 ) ◦ ∪ . . . ∪ S − 1 m ( X \ F n 0 ) ◦ for some S 1 , . . . , S m ∈ G, m ∈ N . Cho ose δ > 0 suc h that if y , z ∈ X with d ( y , z ) < δ then d ( S i ( y ) , S i ( z )) < 1 n 0 for ev ery 1 ≤ i ≤ m. W e claim that if x ∈ S − 1 j ( X \ F n 0 ) ◦ for some 1 ≤ j ≤ m, then x ∈ X \ F 1 δ . Indeed, if there are β ∈ Q \ Z , γ ∈ Z ∗ with dom + ( β ) = ∅ = dom − ( γ ) , max dom − ( β ) < k 1 < k 2 < min dom + ( γ ) su c h that d ( T pβ + q γ i ( x ) , x ) < δ for eve ry 0 ≤ p, q ≤ k , 1 ≤ i ≤ l, then d ( S j ( T pβ + q γ i ( x )) , S j ( x )) = d ( T pβ + q γ i ( S j ( x )) , S j ( x )) < 1 n 0 for ev ery 0 ≤ p, q ≤ k , 1 ≤ i ≤ l , since S j comm utes with { T i,n } n ∈ Z ∗ , 1 ≤ i ≤ l . Since S j ( x ) ∈ ( X \ F n 0 ) ◦ , w e ha v e a con tradiction. Since ev ery x ∈ X is in S − 1 j ( X \ F n 0 ) ◦ for some 1 ≤ j ≤ m, w e ha ve pro v ed t hat x ∈ X \ F 1 δ for ev ery x ∈ X , so, X \ F 1 δ = X a con tradiction according to Theorem 3 .1 .  Definition 4.2. A subset U ⊆ X is called r esidual if it con tains a n en umerable in ter- section o f dense sets. 15 Theorem 4.1 giv es the follow ing. Prop osition 4.3. If ( X, T 1 , . . . , T l ) is a minimal r a tional dynam ic al s ystem, then the set of multiple r e curr ent p oints of X is r esidual. Pr o of . It follo ws from Theorem 4.1, since for ev ery n ∈ N the set F n is dense, op en a nd ∅ 6 = F n ⊆ { m ultiple rational recurren t p oints } .  Prop osition 4.4. The or ems 3.1, 3 .2 and 4.1 ar e e quivalent. Pr o of . W e ha v e already seen that Theorems 3.1 and 3.2 are equiv alent a nd that Theo- rem 4 .1 follows from Theorem 3.1. Let l ∈ N , ( X, T 1 , . . . , T l , R ) be a minimal rational dynamical system, k ∈ N and k 1 < k 2 b e ar bit r a ry real n um b ers. According to the pro of of Theorem 4.1, if U is a non-empty set in X then there exist a x ∈ U and sequences ( β n ) n ∈ N ⊆ Q \ Z , ( γ n ) n ∈ N ⊆ Z ∗ with d om + ( β 1 ) = ∅ = dom − ( γ 1 ) , max dom − ( β 1 ) < k 1 < k 2 < min dom + ( γ 1 ) , β n + γ n ≺ β n +1 + γ n +1 , dom + ( β n ) = ∅ = dom − ( γ n ) for ev ery n ∈ N suc h that T pβ n + q γ n i ( x ) → x for ev ery 0 ≤ p, q ≤ k , 1 ≤ i ≤ l . So, there exists n 0 ∈ N suc h that T pβ n 0 + q γ n 0 i ( x ) ∈ U for ev ery 0 ≤ p, q ≤ k , 1 ≤ i ≤ l. Hence T 0 ≤ p,q ≤ k ( U ∩ ( T pβ n 0 + q γ n 0 1 ) − 1 U ∩ . . . ∩ ( T pβ n 0 + q γ n 0 l ) − 1 U ) 6 = ∅ , the conclusion.  Definition 4.5. Let k ∈ N and k 1 < k 2 arbitrary real n um b ers. A subset A ⊆ Q is called RIP(k,k 1 ,k 2 )-set if there exist sequences ( β n ) n ∈ N ⊆ Q \ Z , ( γ n ) n ∈ N ⊆ Z ∗ with dom + ( β 1 ) = ∅ = dom − ( γ 1 ) , max dom − ( β 1 ) < k 1 < k 2 < min dom + ( γ 1 ) , β n + γ n ≺ β n +1 + γ n +1 , dom + ( β n ) = ∅ = dom − ( γ n ) for ev ery n ∈ N suc h tha t A consists of the n um b ers pβ i + q γ i , 0 ≤ p, q ≤ k together with all the finite sums in the form p ( β i 1 + . . . + β i s ) + q ( γ i 1 + . . . + γ i s ) , 0 ≤ p, q ≤ k with i 1 < . . . < i s . Prop osition 4.6. L et ( X , T 1 , . . . , T l ) a r ational dynamic al system, k ∈ N , k 1 < k 2 arbi- tr ary r e al numb ers a nd x 0 a m ultiple r ational r e c urr ent p oint o f X . Th e n , for every δ > 0 the se t R δ = { q ∈ Q : d ( T q i ( x 0 ) , x 0 ) < δ for every 1 ≤ i ≤ l } c ontains a RIP( k , k 1 , k 2 )-set. Pr o of . Let δ > 0 and x 0 a po in t whic h satisfies the conclusion of Theorem 4.1. According to Theorem 4.1 there exist β 1 ∈ Q \ Z and γ 1 ∈ Z ∗ with dom + ( β 1 ) = ∅ = d om − ( γ 1 ) , max dom − ( β 1 ) < k 1 < k 2 < min dom + ( γ 1 ) suc h that (1) d ( T pβ 1 + q γ 1 i ( x 0 ) , x 0 ) < δ for ev ery 0 ≤ p, q ≤ k , 1 ≤ i ≤ l. Let 0 < δ 2 ≤ δ suc h that (2) d ( x, x 0 ) < δ 2 ⇒ d ( T pβ 1 + q γ 1 i ( x ) , x 0 ) < δ for ev ery 0 ≤ p, q ≤ k , 1 ≤ i ≤ l . 16 According to Theorem 4.1 there exist β 2 ∈ Q \ Z and γ 2 ∈ Z ∗ with β 1 + γ 1 ≺ β 2 + γ 2 , dom + ( β 2 ) = ∅ = dom − ( γ 2 ) suc h that (3) d ( T pβ 2 + q γ 2 i ( x 0 ) , x 0 ) < δ 2 for ev ery 0 ≤ p, q ≤ k , 1 ≤ i ≤ l. The conditions (1) , (2) and (3) ensures that ( ∗ ) d ( T m i ( x 0 ) , x 0 ) < δ , where m = pβ 1 + q γ 1 or pβ 2 + q γ 2 or p ( β 1 + β 2 ) + q ( γ 1 + γ 2 ) fo r ev ery 0 ≤ p, q ≤ k , 1 ≤ i ≤ l. Assume that w e ha v e fo und β 1 , . . . , β n , γ 1 , . . . , γ n with β s + γ s ≺ β s +1 + γ s +1 for ev ery s = 1 , . . . , n − 1 , dom + ( β s ) = ∅ = d om − ( γ s ) , 1 ≤ s ≤ n suc h that ( ∗ ) ho lds for m = p ( β i 1 + . . . + β i s ) + q ( γ i 1 + . . . + γ i s ) for ev ery 0 ≤ p, q ≤ k , 1 ≤ i ≤ l with i 1 < . . . < i s ≤ n. Let δ n +1 ≤ δ suc h that d ( x, x 0 ) < δ n +1 ⇒ d ( T m i ( x ) , x 0 ) < δ for the previous m, 1 ≤ i ≤ l . According to Theorem 4 .1 there exist β n +1 ∈ Q \ Z a nd γ n +1 ∈ Z ∗ with β n + γ n ≺ β n +1 + γ n +1 , dom + ( β n +1 ) = ∅ = dom − ( γ n +1 ) such that d ( T pβ n +1 + q γ n +1 i ( x 0 ) , x 0 ) < δ n +1 for ev ery 0 ≤ p, q ≤ k , 1 ≤ i ≤ l . Then, ( ∗ ) holds if w e replace m with m + pβ n +1 + q γ n +1 or pβ n +1 + q γ n +1 for ev ery 0 ≤ p, q ≤ k . Inductiv ely , w e hav e that R = { p ( β i 1 + . . . + β i s ) + q ( γ i 1 + . . . + γ i s ) , 0 ≤ p, q ≤ k with i 1 < . . . < i s } ⊆ R δ .  5. Some Ap plica tions In this last section w e will presen t some applications of Theorem 3 .3 not only to top ology but also to diophantine approximations and n um b er theory . A finite partitio n of Q l , l ∈ N can b e considered as a function from Q l to a finite set. Analogously to [F u] (Theorem 2 .9 and Lemma 2.11) w e extend Theorem 3.3 from the finite partitions of Q l to functions from Q l to a compact metric space. Theorem 5.1. L et l ∈ N , k 1 < k 2 b e arbitr ary r e al numb ers a n d let f : Q l → X b e an arbitr ary function w i th values on the c om p act metric sp ac e X . F or any ε > 0 , k ∈ N an d F ∈ [ Q l ] <ω > 0 we c an find α ∈ Q l , β ∈ Q \ Z and γ ∈ Z ∗ with dom + ( β ) = ∅ = d om − ( γ ) , max dom − ( β ) < k 1 < k 2 < min dom + ( γ ) such that diam ( f ( α + ( pβ + q γ ) F )) < ε for every 0 ≤ p, q ≤ k . Pr o of . Let X = S r i =1 U i , r ∈ N where diam ( U i ) < ε for ev ery 1 ≤ i ≤ r . Then Q l = S r i =1 f − 1 ( U i ) . W e obtain the result applying Theorem 3.3 to this partition.  W e will no w giv e an application of Theorem 5.1 to diophantine approximations. Let δ b e an arbitrary real n um b er and f ( q ) = e iπ q 2 δ , q ∈ Q . According to Theorem 5.1, for ε > 0 , k ∈ N and k 1 < k 2 arbitrary real n um b ers there exist α ∈ Q , β ∈ Q \ Z and 17 γ ∈ Z ∗ with d om + ( β ) = ∅ = dom − ( γ ) , max dom − ( β ) < k 1 < k 2 < min dom + ( γ ) suc h that | f ( α ) − f ( α + ( pβ + q γ )) | < ε and | f ( α ) − f ( α + 2( pβ + q γ )) | < ε for ev ery 0 ≤ p, q ≤ k . If h ≡ h ( p, q ) = pβ + q γ , we hav e | 1 − e i (2 αh + h 2 ) πδ | < ε a nd | 1 − e i (4 αh +4 h 2 ) πδ | < ε. Since cos( x ) ≤ 1 for eve ry x, we hav e | 1 − e 2 ix | ≤ 2 | 1 − e ix | . 2 h 2 π δ = [(4 αh + 4 h 2 ) − 2(2 αh + h 2 )] π δ, so | 1 − e 2 ih 2 π δ | ≤ 2 | 1 − e i (2 αh + h 2 ) πδ | + | 1 − e i (4 αh +4 h 2 ) πδ | < 3 ε. If we set ξ ≡ ξ ( p, q ) = h 2 δ − [ h 2 δ ] fo r ev ery 0 ≤ p, q ≤ k , we ha v e | 1 − e 2 π iξ | < 3 ε ⇔ 2 sin( π ξ ) < 3 ε. Using the inequalit y sin( x ) ≥ 2 π x for 0 ≤ x ≤ π 2 , w e get: ( i ) If π ξ ∈ [0 , π 2 ] , then for m ≡ m ( p, q ) = [ h 2 δ ] we hav e | h 2 δ − m | < ε. ( ii ) If π ξ ∈ ( π 2 , π ) then π − π ξ ∈ (0 , π 2 ) , so, for m ≡ m ( p, q ) = [ h 2 δ ] + 1 w e ha ve | h 2 δ − m | < ε. This implies that there exists m ( p, q ) ∈ Z suc h that | δ ( pβ + q γ ) 2 − m ( p, q ) | < ε for ev ery 0 ≤ p, q ≤ k . Note that (for p = 0) w e hav e tha t for ev ery δ real num b er we can solv e | δ n 2 − m | < ε for ev ery ε > 0 (a result first prov ed b y Hardy and L it t lewoo d). W e will no w state and pro v e the m ultidimensional v ersion of Theorem 5.1. Theorem 5.2. L et l 1 , . . . , l s ∈ N , s ∈ N and k 1 < k 2 b e arbi tr ary r e al numb ers. L et f 1 : Q l 1 → X 1 , . . . , f s : Q l s → X s b e s arbitr ary functions with values on the c omp act m e tric sp ac es X 1 , . . . , X s r esp e ctively. F or any ε > 0 , k ∈ N and F 1 ∈ [ Q l 1 ] <ω > 0 , . . . , F s ∈ [ Q l s ] <ω > 0 we c an find α i ∈ Q l i , 1 ≤ i ≤ s, β ∈ Q \ Z and γ ∈ Z ∗ with dom + ( β ) = ∅ = dom − ( γ ) , max dom − ( β ) < k 1 < k 2 < min dom + ( γ ) such that diam ( f i ( α i + ( pβ + q γ ) F i )) < ε for every 0 ≤ p, q ≤ k , 1 ≤ i ≤ s. Pr o of . F o rm f 1 × . . . × f s : Q l 1 + ... + l s → X 1 × . . . × X s , F = F 1 × . . . × F s and apply Theorem 5.1 .  Let giv e an application of Theorem 5.2 to n um b er theory . Let π ( x ) b e a real p olynomial with π (0) = 0 , k ∈ N and k 1 < k 2 arbitrary real n um b ers. W e will show that for ev ery ε > 0 t here exist β ∈ Q \ Z , γ ∈ Z ∗ with dom + ( β ) = ∅ = dom − ( γ ) , max dom − ( β ) < k 1 < k 2 < min dom + ( γ ) and in tegers m ( p, q ) , 0 ≤ p, q ≤ k suc h tha t | π ( pβ + q γ ) − m ( p, q ) | < ε for ev ery 0 ≤ p, q ≤ k . Let π ( x ) = b n x n + . . . + b 1 x. W e write π ( x ) = s n A n x n + . . . + s 1 A 1 x, with A r = P r j =0 ( − 1) j  r j  j r , r = 1 , . . . , n (where  r j  = r ! j !( r − j )! ). 18 F or ev ery r = 1 , . . . , n we set f r : Q → R / Z with f r ( q ) = s r q r . According to Theo- rem 5.2 there exist α 1 , . . . , α n ∈ Q , β ∈ Q \ Z , γ ∈ Z ∗ with dom + ( β ) = ∅ = d om − ( γ ) , max dom − ( β ) < k 1 < k 2 < min dom + ( γ ) suc h that k f r ( α r + j ( pβ + q γ )) − f r ( α r ) k < ε 2 n +1 , j = 1 , . . . , r , 0 ≤ p, q ≤ k . W e can easily prov e by induction that P r j =0 ( − 1) j  r j  ( x + j y ) r = A r y r , so, w e ha v e that P r j =0 ( − 1) j  r j  f r ( α r + j ( pβ + q γ )) = A r f r ( pβ + q γ ) fo r ev ery 0 ≤ p, q ≤ k . Since, P r j =0 ( − 1) j  r j  f r ( α r ) = 0 and P r j =0  r j  = 2 r , w e hav e that k A r f r ( pβ + q γ ) k < 2 r ε 2 n +1 , r = 1 , . . . , n, thus k π ( pβ + q γ ) k < ε 2 n +1 ( P n r =1 2 r ) < ε fo r ev ery 0 ≤ p, q ≤ k . Note that (for p = 0) for ev ery real p olynomial π ( x ) with π (0) = 0 and ε > 0 , w e can find in tegers m, n with | π ( n ) − m | < ε (a result first prov ed b y Hardy and W eyl). Ac kno wledgmen t s. The author wish to thank Prof es sors V. F armaki and S. Negrep on- tis for helpful discuss ions and supp ort during the preparatio n of this pap er. T he author also ackno wledge partial supp ort f r om the State Sc holarship F oundation of Greece. Reference s [Bi] G. D. Birk hoff, Dynamic al Systems , Amer. Math. Soc . Collo q. 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Wisk. 15 (192 7 ), 212–2 16. Andreas K outsogiannis: Dep ar tment of Ma thema tics, A th ens University, P anepistemiopolis, 15784 A thens, Greece E-mail address: akoutsos@math.uo a .gr 19