On the continuity of the elements of the Ellis semigroup and other properties
aa r X i v : . [ m a t h . GN ] S e p ON THE CONTINUITY OF THE ELEMENTS OF THE ELLIS SEMIGROUPAND OTHER PROPERTIES
S. GARC´IA-FERREIRA, Y. RODR´IGUEZ-L ´OPEZ, AND C. UZC ´ATEGUI
Abstract.
We consider discrete dynamical systems whose phase spaces are compact metrizablecountable spaces. In the first part of the article, we study some properties that guarantee thecontinuity of all functions of the corresponding Ellis semigroup. For instance, if every accumulationpoint of X is fixed, we give a necessary and sufficient condition on a point a ∈ X ′ in order that allfunctions of the Ellis semigroup E ( X, f ) be continuous at the given point a . In the second part,we consider transitive dynamical systems. We show that if ( X, f ) is a transitive dynamical systemand either every function of E ( X, f ) is continuous or | ω f ( x ) | = 1 for each accumulation point x of X , then E ( X, f ) is homeomorphic to X . Several examples are given to illustrate our results. Introduction and preliminaries
A dynamical systems (
X, f ) will consist of a compact metric space X and a continuous function f : X → X (usually, this kind of dynamical systems are called discrete dynamical systems and weshort the name for convenience). The orbit of point x ∈ X is the set O f ( x ) := { f n ( x ) : n ∈ N } .The system ( X, f ) is transitive if there is a point with dense orbit. A point x ∈ X is called periodic if there is n ≥ f n ( x ) = x , and its period is min { n ∈ N : f n ( x ) = x } . The symbol P f stands for the set of all periods of the periodic points of a dynamical systems ( X, f ). A point x iscalled eventually periodic if its orbit is finite. The ω − limit set of x ∈ X , denoted by ω f ( x ), is theset of points y ∈ X for which there exists an increasing sequence ( n k ) k ∈ N such that f n k ( x ) → y .Observe that for each y ∈ O f ( x ), we have that ω f ( y ) = ω f ( x ). If O f ( y ) contains a periodicpoint x , then ω f ( y ) = O f ( x ). For a space X we denote by N ( x ) the set of all neighborhoods of x ∈ X , and the set of all accumulation points of X will be simply denoted by X ′ . The Stone- ˇCechcompactification β ( N ) of N with the discrete topology will be identified with the set of ultrafiltersover N . Its remainder is denoted by N ∗ = β ( N ) \ N is the set of all free ultrafilters on N , where,as usual, each natural number n is identified with the fixed ultrafilter consisting of all subsets of N containing n .Since our phase spaces are compact metric countable spaces, we remind the reader the classicalresult from [12] which asserts that every compact metric countable space is homeomorphic to acountable sucesor ordinal with the order topology. In this context, some of the most attractivephase space has the form ω α + 1, where α ≥ Ellis semigroup of a dynamical system (
X, f ), denoted E ( X, f ), is defined as the pointwiseclosure of { f n : n ∈ N } in the compact space X X with composition of functions as its algebraicoperation. The Ellis semigroup is equipped with the topology inhered from the product space X X . This semigroup was introduced by R. Ellis in [3], and it is a very important tool in the studyof the topological behavior of a dynamical systems. The article [10] offers an excellent survey
Mathematics Subject Classification.
Primary 54H20, 54G20: secondary 54D80.
Key words and phrases. discrete dynamical system, Ellis semigroup, p -iterate, p -limit point, ultrafilter, compactmetric countable space.Research of the first-named author was supported by the PAPIIT grant no. IN-105318. The third author wassupported by the VIE-UIS grant oncerning applications of the Ellis semigroup. In the paper [9], the authors initiated the study ofthe continuity and discontinuity of the elements of E ( X, f ) \ { f n : n ∈ N } . For instance, they pointout that if X is a convergent sequence with its limit point, then all the elements of E ( X, f ) areeither continuous or discontinuous (this result was later improved in [7]). In a different context, P.Szuca [13] showed that if X = [0 , f : [0 , → [0 ,
1] is onto and f p is continuous forsome p ∈ N ∗ , then all the elements of E ([0 , , f ) are continuous. Using the Cantor set as a phasespace and a generalized ship maps, the continuity and discontinuity of the elements of the Ellissemigroup where studied in [6]. The main tool that have been used in all these investigations is thecombinatorial properties of the ultrafilters on N . Certainly, the Ellis semigroup can be describedin terms of the notion of convergence with respect to an ultrafilter. Indeed, given p ∈ N ∗ and asequence ( x n ) n ∈ N in a space X , we say that a point x ∈ X is the p − limit point of the sequence, insymbols x = p − lim n →∞ x n , if for every neighborhood V of x , { n ∈ N : f n ( x ) ∈ V } ∈ p . Observethat a point x ∈ X is an accumulation point of a countable set { x n : n ∈ N } of X iff there is p ∈ N ∗ such that x = p − lim n →∞ x n .The notion of a p − limit point has been used in several branches of mathematics (see for instance[2] and [4, p. 179]). A. Blass [1] and N. Hindman [11] formally established the connection between“the iteration in topological dynamics” and “the convergence with respect to an ultrafilter” byconsidering a more general iteration of the function f as follows: Let X be a compact space and f : X → X a continuous function. For p ∈ N ∗ , the p − iterate of f is the function f p : X → X defined by f p ( x ) = p − lim n →∞ f n ( x ) , for each x ∈ X . The description of the Ellis semigroup and its operation in terms of the p − iteratesare the following (see [1], [11]): E ( X, f ) = { f p : p ∈ β N } f p ◦ f q = f q + p for each p, q ∈ β N . We will use the following notation E ( X, f ) ∗ := E ( X, f ) \ { f n : n ∈ N } . Besides, we have that ω f ( x ) = { f p ( x ) : p ∈ N ∗ } for each x ∈ X .Recently (for instance see [7] and [8]), we have investigated the structure of the Ellis semigroupof a dynamical system and the topological properties of some of its elements. Our main purposemoves in two directions: The first one concerns with the continuity and discontinuity of the p -iterates which is dealt in the third section, and the second one concerns about a general questionstated in [8]: Question 1.1.
Given two compact metric countable spaces X and Y, is there a continuous function f : X → X such that E ( X, f ) is homeomorphic to Y ? We provide a partial answer to this question in the forth section.2.
Basic Properties
In this section, we state several useful results that were proved in the articles [7] and [8]. Ourfirst lemma is precisely Lemma 2.1 from [8].
Lemma 2.1.
Let ( X, f ) be a dynamical system and x ∈ X . ( i ) Assume that x is periodic with period n and let l < n . Then, p ∈ (cid:0) n N + l (cid:1) ∗ iff f p ( x ) = f l ( x ) . ii ) Suppose that x is eventually periodic and that m ∈ N is the smallest positive integer suchthat f m ( x ) is a periodic point. If n is the period of f m ( x ) and p ∈ (cid:0) n N + l (cid:1) ∗ for some l < n ,then f p ( x ) = f l ( f nj ( x )) where j = min { i : m ≤ ni + l } . ( iii ) Suppose that the orbit of x is infinite and ω f ( x ) = O f ( y ) for some periodic point y ∈ X with period n . If p, q ∈ ( n N + l ) ∗ for some l < n , then f p ( x ) = f q ( x ) . ( iv ) f p ( f n ( x )) = f n ( f p ( x )) for every n ∈ N , x ∈ X and every p ∈ N ∗ . The next statement is Lemma 2.4 of [7].
Lemma 2.2.
Let ( X, f ) be a dynamical system. If ω f ( x ) is finite, then every point of ω f ( x ) isperiodic. In particular, if ω f ( x ) has an isolated point in O f ( x ) , then every point of ω f ( x ) is periodic. The following lemma is a reformulation of Theorem 2.2 from [8].
Lemma 2.3.
Let ( X, f ) be a dynamical system, x ∈ X and m ∈ N . Then |O f ( x ) | > m iff f m ( x ) = f n ( x ) for every m = n . In particular, there is an integer M > such that |O f ( x ) | < M for each x ∈ X iff E ( X, f ) is finite. We omit the proof of the following well-known result.
Lemma 2.4.
Let ( X, f ) be a dynamical system such that X has a dense subset consisting ofisolated points. If X has a point with infinite orbit, then { f n : n ∈ N } is infinite and discrete in E ( X, f ) and f n = f p for every ( n, p ) ∈ N × N ∗ . In addition, if the orbit of w is dense in X , then X = { f p ( w ) : p ∈ β ( N ) } and thus X is a continuous image of β ( N ) . Continuity of the p -iterates In the context of compact metric countable spaces, we showed in [7, Th. 3.11] that if all pointsof a ∈ X ′ are periodic, then for b ∈ X ′ either each f q is discontinuous at b for every q ∈ N ∗ or each f q is continuous at b for all q ∈ N ∗ . An example where this assertion fails assuming that all pointsof X ′ are eventually periodic is also given in [7]. Here, our main task is to give a necessary andsufficient condition on the space X in order that all p -iterates be discontinuous at a given point.To have this done we shall need some preliminary lemmas.The next two results correspond to Lemma 3.7 and Theorem 3.8 of [7], respectively. Lemma 3.1.
Let ( X, f ) be a dynamical system such that X is a compact metric countable spaceand every point of X ′ is periodic. If x has an infinite orbit and y ∈ ω f ( x ) is fixed, then f n ( x ) −→ y . Lemma 3.2.
Let ( X, f ) be a dynamical system such that X is a compact metric countable spaceand every point of X ′ is periodic. For every x ∈ X , there exists a periodic point y ∈ X such that ω f ( x ) = O f ( y ) . The following two additional results are needed to establish our theorems.
Lemma 3.3.
Let f : X → X be a continuous function such that every accumulation point is fixed.If V , W are nonempty open sets such that V ∩ W = ∅ , then the set { x ∈ X : x ∈ V and f ( x ) ∈ W } is finite.Proof. Assume, towards a contradiction, that H = { x ∈ X : x ∈ V y f ( x ) ∈ W } is infinite. Since X is compact and metric, there is a non constant sequence ( a n ) n ∈ N in H and a ∈ V ∩ X ′ such that a n → a . As f is continuous and a is fixed, then f ( a n ) → f ( a ) = a which implies that a ∈ W . Butthis is a contradiction. This shows that H is finite. (cid:3) Lemma 3.4.
Let ( X, f ) be a dynamical system such that X is a compact metric countable spaceand every accumulation point is periodic. Let p ∈ N ∗ and b ∈ X be an isolated point. If there is asequence ( a n ) n ∈ N in X such that f p ( a n ) → b , then b is periodic and O f ( b ) = ω f ( b ) = ω f ( a n ) forevery positive integer except finitely many. roof. It follows directly from the assumption that B = { n ∈ N : f p ( a n ) = b } is cofinite andhence b ∈ ω f ( a n ) ⊆ O f ( a n ) for every n ∈ B . By Lemma 2.2, we have that b is a periodic point.So, O f ( b ) = ω f ( b ). By Lemma 3.2, for every n ∈ B there is a periodic point y n ∈ X such that ω f ( a n ) = O f ( y n ) and since b ∈ O f ( y n ), we conclude that ω f ( b ) = O f ( b ) = O f ( y n ) = ω f ( a n ) for all n ∈ B . (cid:3) Theorem 3.5.
Let ( X, f ) be a dynamical system such that X is a compact metric countable spaceand every accumulation point of X is fixed. For every a ∈ X ′ , the following statements are equiv-alent: (1) There is p ∈ N ∗ such that f p is discontinuous at a ∈ X ′ . (2) There are a periodic point b ∈ X \ { a } and a sequence ( a n ) n ∈ N in X such that a n → a and b ∈ O f ( a n ) for all n ∈ N . (3) f p is discontinuous at a , for each p ∈ N ∗ .Proof. (1) ⇒ (2). Suppose that f p is discontinuous at a ∈ X ′ . Then, there is a nontrivial sequence( a n ) n ∈ N in X such that a n → a and f p ( a n ) does not converge to a . Since X is compact andmetric, there are a sequence of positive integers ( n k ) k ∈ N and b ∈ X \ { a } such that f p ( a n k ) → b .Assume, without loss of generality, that this subsequence is ( a n ) n ∈ N . In virtue of Lemma 3.4, weonly consider the case when b ∈ X ′ . Since a = b , f p ( a n ) → b and a n → a , there are a clopenset V ∈ N ( a ) and N ∈ N such that b / ∈ V , a n ∈ V y f p ( a n ) ∈ X \ V for each n ≥ N . Forevery n ≥ N there is d n ∈ O f ( a n ) ∩ V such that f ( d n ) ∈ X \ V , this is possible since the set A n = { m ∈ N : f m ( a n ) / ∈ V } ∈ p and hence it is infinite. Then, by Lemma 3.3, we have that theset B = { d n : n ≥ N } is finite. Hence, there exists d ∈ B for which the set H = { n ∈ N : d = d n } is infinite. To finish the proof it suffices to show that b ∈ ω f ( a n ) for all n ∈ H . Indeed, we considertwo cases. Suppose first that O f ( d ) is infinite, then there is e ∈ X ′ such that e ∈ ω f ( d ). As e is fixed, by Lemma 3.1, f m ( d ) → e . Analogously it is shown that f m ( a n ) −−−−→ m →∞ e because of e ∈ ω f ( a n ), for each n ∈ H . Consequently, we obtain that f q ( a n ) = e for each q ∈ N ∗ and for eachall n ∈ H . This implies that b = e since f p ( a n ) −−−→ n ∈ H b . Thus b ∈ ω f ( a n ) for all n ∈ H . For thesecond case, we assume that O f ( d ) is finite. As d ∈ O f ( a n ) for each n ∈ H , we also have that theorbit of a n is finite for all n ∈ H . By Lemma 3.2, we may choose a periodic point e ∈ X such that O f ( e ) = ω f ( d ) = { f q ( d ) : q ∈ N ∗ } = { f q ( a n ) : q ∈ N ∗ } = ω f ( a n ) for n ∈ H . Since O f ( e ) is finiteand f p ( a n ) → b , we obtain that b ∈ ω f ( a n ), for each n ∈ H .(2) ⇒ (3). We have to analyze two possible cases.Caso I. Suppose b is isolated. By assumption, b is periodic. Hence, if b ∈ O f ( y ), then O f ( y )is finite and so O f ( b ) = ω f ( y ) = { f p ( y ) : p ∈ N ∗ } . Thus, from the hypothesis we obtain that f p ( a n ) ∈ ω f ( a n ) = O f ( b ) for every n ∈ N and every p ∈ N ∗ . As a is fixed and b = a , then a
6∈ O f ( b )and the sequence ( f p ( a n )) n ∈ N cannot converge to a for any p ∈ N ∗ . Thus f is discontinuous at a .Caso II. Suppose b ∈ X ′ . Observe that if b ∈ O f ( y ) and O f ( y ) is finite, then ω f ( y ) = { b } . Onthe other hand, if b ∈ O f ( y ) and O f ( y ) is infinite, then it follows from Lemma 3.1 that ω f ( y ) = { b } .Thus, we have that f p ( a n ) = b for each p ∈ N ∗ and for each n ∈ N . Thus f is discontinuous at a .(3) ⇒ (1). It is evident. (cid:3) Corollary 3.6.
Let ( X, f ) be a dynamical system such that X is a compact metric countable spaceand every accumulation point of X is fixed. If the orbit of every isolated point of X is finite, thenevery function of E ( X, f ) is continuous.Proof. Assume that there is p ∈ N ∗ such that f p is discontinuous at a ∈ X ′ . It follows fromTheorem 3.5(2) that there are a periodic point b ∈ X \ { a } and a sequence ( a n ) n ∈ N in X suchthat a n → a and b ∈ O f ( a n ) for all n ∈ N . Since each accumulation point of X is fixed, we may ssume that a n is isolated for every n ∈ N ∗ , hence b ∈ O f ( a n ) for all n ∈ N . Now, let V and W be disjoint open sets such that V ∩ W = ∅ , a ∈ V and b ∈ W . Without loss of generality, we mayassume that a n ∈ V for all n ∈ N . Then we can find a sequence ( k n ) n ∈ N such that f k n ( a n ) ∈ V and f k n +1 ( a n ) ∈ W for every n ∈ N . We may assume that f k n ( a n ) → c ∈ V , but this is impossiblesince f k n +1 ( a n ) → f ( c ) = c ∈ W . Therefore, f p is continuous. (cid:3) The conclusion of Theorem 3.5 is not true if we replace the hypothesis “every accumulation pointis fixed” by the hypothesis “every accumulation point is periodic”. Indeed, the next two exampleswitness that the condition (2) of Theorem 3.5 holds together with either the assumption “ f p iscontinuous for every p ∈ N ∗ ” or the assumption “ f p is discontinuous for all p ∈ N ∗ ” .The phase space of the following dynamical systems is the ordinal space 2 ω + 1 (two disjointconvergent sequences) which will be identified with the following subspace of R : X = { a n : n ∈ N } ∪ { b n : n ∈ N } ∪ { a, b } , where a n < a n +1 < a < b n < b n +1 < b for every n ∈ N , a n → a and b n → b . Example 3.7.
Define a function f : X → X as follows: a ) f ( a ) = b and f ( b ) = a , b ) f ( a n ) = b n for each n ∈ N , and c ) f ( b n ) = a n +1 for each n ∈ N .That is, a → b → a → b → a → b → a → b · · · a n → b n → a n +1 → b n +1 → a n +2 → b n +2 → · · · It is evident that f is continuous. From the definition of f we have, for all n, m ∈ N , the following: i ) f m ( a n ) = a n + m , ii ) f m +1 ( a n ) = b n + m , iii ) f m ( b n ) = b n + m , and iv ) f m +1 ( b n ) = a n + m +1 .Conditions i ) − iv ) imply the following: (1) f p ( a n ) = f p ( a ) = a for every p ∈ (2 N ) ∗ , (2) f p ( a n ) = f p ( b ) = b for every p ∈ (2 N + 1) ∗ , (3) f p ( b n ) = f p ( b ) = b for every p ∈ (2 N ) ∗ , and (4) f p ( b n ) = f p ( a ) = a for every p ∈ (2 N + 1) ∗ .Then we have that the accumulation points a and b have period equal to and both satisfy thesecond condition of Theorem 3.5. However, the function f p is continuous for every p ∈ N ∗ . Example 3.8.
We define a function f : X → X as follows: a ) f ( a ) = b and f ( b ) = a , b ) f ( a ) = a , c ) f ( a n ) = b n − for every < n ∈ N , and d ) f ( b n ) = a n , for every n ∈ N .That is, b n → a n → b n − → a n − → b n − → a n − → b n − → a n − → · · · b → a → b → a → b → a . It is not hard to prove that f is continuous. From the definition of f we easily have that for each x ∈ X \ { a, b } there exists n ∈ N such that f n ( x ) = a . Hence, f p ( x ) = a for every x ∈ X \ { a, b } and every p ∈ N ∗ . Since f p ( a ) = a and f p ( b ) = b for all p ∈ (2 N ) ∗ ; and f p ( a ) = b and f p ( b ) = a for all p ∈ (2 N + 1) ∗ , we conclude that f p is discontinuous at a and b for each p ∈ N ∗ . Finally,observe that a ∈ O f ( a n ) for all n ∈ N , so (2) of Theorem 3.5 is satisfied. n the next theorem, we show a generalization of (2) ⇒ (3) in Theorem 3.5. Theorem 3.9.
Let ( X, f ) be a dynamical system such that X is a compact metric countable spaceand every accumulation point of X is periodic. Let a ∈ X ′ . If there are a sequence ( a n ) n ∈ N in X and a periodic point b ∈ X \ O f ( a ) such that a n → a and b ∈ O f ( a n ) for every n ∈ N , then f p isdiscontinuous at a for each p ∈ N ∗ .Proof. First we show a particular case. Suppose that b ∈ O f ( a n ) for infinitely many n ∈ N . Hence,from the periodicity of b , we have that O f ( b ) = ω f ( b ) = ω f ( a n ) = { f p ( a n ) : p ∈ N ∗ } is finite forinfinitely many n ∈ N . Since O f ( a ) ∩ O f ( b ) = ∅ , the sequence ( f p ( a n )) n ∈ N cannot converge to f p ( a ) ∈ O f ( a ) for any p ∈ N ∗ . Thus f p is not continuous at a for any p ∈ N ∗ .For the proof of the theorem we consider two cases: (i) Suppose b is isolated. Then b ∈ O f ( a n )for every n ∈ N and we are done from the result we proved above.(ii) Assume that b ∈ X ′ \ (cid:0) S n ∈ N O f ( a n ) (cid:1) . From the result proved above, we can assume that O f ( a n ) is infinite for every n ∈ N . Then, we must have that b ∈ ω f ( a n ) for every n ∈ N . Now, invirtue of Lemma 3.2, for every n ∈ N there is a periodic point y n ∈ X such that ω f ( a n ) = O f ( y n ).Since b is periodic and b ∈ O f ( y n ), then O f ( b ) = O f ( y n ) for all n ∈ N . Thus f p ( a n ) ∈ O f ( b ) forall n and all p ∈ N ∗ . Since O f ( a ) ∩ O f ( b ) = ∅ , we conclude as before that f p is not continuous at a for any p ∈ N ∗ . (cid:3) In the next result we slightly modify the proof of the previous theorem to get an interestingstatement.
Theorem 3.10.
Let ( X, f ) be a dynamical system such that X is a compact metric countable spaceand every accumulation point of X is periodic. Let a ∈ X ′ and ( a n ) n ∈ N be a sequence in X suchthat f p ( a n ) → f p ( a ) , for some p ∈ N ∗ . Suppose b ∈ X is a periodic point and b ∈ T n ∈ N O f ( a n ) ,then b ∈ O f ( a ) .Proof. Let a ∈ X ′ , ( a n ) n in X and p ∈ N ∗ as in the hypothesis. First, suppose that b is isolated.Then, b ∈ T n ∈ N O f ( a n ) and so O f ( b ) = ω f ( b ) = ω f ( a n ) for every n ∈ N . Since f p ( a n ) → f p ( a ) ∈O f ( a ) and f p ( a n ) ∈ ω f ( a n ) = O f ( b ) for each n ∈ N , we must have that O f ( a ) ∩ O f ( b ) = ∅ andso b ∈ O f ( a ). Now, suppose that b ∈ X ′ . Notice that if b ∈ O f ( a n ) for some n ∈ N , then O f ( b ) = ω f ( b ) = ω f ( a n ). Thus we have that b ∈ ω f ( a n ) for all n ∈ N . By Lemma 3.2, there existsa periodic point y n ∈ X such that ω f ( a n ) = O f ( y n ). Since b ∈ ω f ( a n ), then ω f ( a n ) = O f ( b ) forall n ∈ N . Thus, we have shown that f p ( a n ) ∈ O f ( b ) for every n ∈ N . Then, as before, we have O f ( a ) ∩ O f ( b ) = ∅ and thus b ∈ O f ( a ). (cid:3) Concerning the last theorem we have the following example.
Example 3.11.
We consider again the countable ordinal space ω + 1 identified with the subspace X of R from above. Define the function f : X → X as follows: a ) f ( a ) = b and f ( b ) = a , b ) f ( a ) = b , c ) f ( a n ) = b n − for each n > , and d ) f ( b n ) = a n for each n ∈ N .The function f is evidently continuous. From the definition we can see that b n → a n → b n − → a n − → b n − → a n − → · · · b → a → b → a → b → a → b. Hence, all points are eventually periodic. Besides, we have the following properties: i ) For every x ∈ X there is n ∈ N such that f n ( x ) = b . ii ) f p ( x ) = b for each x ∈ X \ { a, b } and for each p ∈ N ∗ . iii ) f p ( a ) = a for all p ∈ (2 N ) ∗ . v ) f p ( a ) = b for all p ∈ (2 N + 1) ∗ . v ) f p ( b ) = a for all p ∈ (2 N + 1) ∗ .Thus, we have that f p is discontinuous at a for all p ∈ (2 N ) ∗ and we also have that b ∈ O f ( x ) forevery x ∈ X . Moreover, f p is discontinuous at b for all p ∈ (2 N + 1) ∗ Theorem 3.5 suggests the following problem.
Problem 3.12.
Let ( X, f ) be a dynamical system such that X is a compact metric countable spacesuch that every accumulation point of X is periodic. Find a necessary and sufficient condition ona point a ∈ X , like in Theorem 3.5, in order that f p is discontinuous at a for each p ∈ N ∗ . Transitive dynamical systems
Continuing the work presented in [8], in this section we focus our attention on transitive dy-namical systems. We recall some questions from that paper. The first one is whether E ( X, f ) iscountable for every transitive system (
X, f ) (see Questions 4.6 in [8]). For instance, this happenswhen every function in E ( X, f ) is continuous (see [8, Theorem 2.9, Theorem 3.3]). Below we ex-tend this result. The second question is whether or not is there a continuous function f : X → X such that E ( X, f ) is homeomorphic to Y , where X and Y are arbitrary compact metric countablespaces? (see Questions 4.8 in [8]).We need the following result from [8] (see Lemma 3.1 and Theorem 2.3). Lemma 4.1.
Let ( X, f ) be a dynamical system.(i) E ( X, f ) \ { f n : n ∈ N } is finite iff there is m ∈ N such that | ω f ( x ) | ≤ m for each x ∈ X .(ii) If ( X, f ) is transitive and y ∈ X ′ , then f ( y ) ∈ X ′ .(iii) If w has a dense orbit, then w is isolated. Theorem 4.2.
Let ( X, f ) be a transitive dynamical system where X is a compact metrizable count-able space. If either f p is continuous for each p ∈ N ∗ , or | ω f ( x ) | = 1 for each x ∈ X ′ , then E ( X, f ) is homeomorphic to X .Proof. Let w be a point of X whose orbit is dense in X . According to Lemma 4.1 the point w must be isolated. Now, consider the function h : E ( X, f ) → X defined by h ( f p ) = f p ( w ), for every p ∈ β ( N ). This function is continuous since it is the restriction of the projection map π w : X X → X to E ( X, f ). It follows from Lemma 2.4 that h is surjective since X = { f p ( w ) : p ∈ β ( N ) } . To provethe theorem it suffices to show that the function h is injective. To have this done, first observe that f p ( f n ( w )) = f n ( f p ( w )), for all n ∈ N and for all p ∈ β ( N ), and the orbit of w is the collection ofisolated points of X . Hence, we obtain that f p ( x ) = f q ( x ) for every isolated point x ∈ X whenever f p ( w ) = f q ( w ) for some p, q ∈ β ( N ).Suppose first that f p is continuous, for each p ∈ N ∗ . Then, if p, q ∈ N ∗ and f p and f q agree onall the points of a dense orbit, then we obtain that f p = f q . This shows h is injective.Now, assume that | ω f ( x ) | = 1 for each x ∈ X ′ . Then for every x ∈ X ′ there is z x ∈ X ′ such that ω f ( x ) = { z x } and hence we obtain that f p ( x ) = z x for all p ∈ N ∗ . Thus we have that f p = f q iff f p ( w ) = f q ( w ) for all p, q ∈ β ( N ). So, h is injective.Therefore, in both cases, h is a homeomorphism between E ( X, f ) and X . (cid:3) Question 4.8 is related to the second condition of the previous theorem.
Corollary 4.3.
Let ( X, f ) be a transitive dynamical system where X is a compact metrizablecountable space. If f p is continuous for each p ∈ N ∗ , then the set P f is finite.Proof. By Theorem 4.2, we know that E ( X, f ) is countable. If P f were infinite, by Theorem 2.7 of[8], then E ( X, f ) would be homeomorphic to the Cantor set 2 N . So, P f is finite. (cid:3) s we have seen in the proof of the previous corollary, if X is a compact metrizable space and E ( X, f ) is countable, then P f must be finite. Next we shall estimate the cardinality of the Ellissemigroup under some restriction on the ω -sets. Theorem 4.4.
Let ( X, f ) be a transitive dynamical system such that X is a compact metric coun-table space. If there is m ∈ N such that | ω f ( x ) | ≤ m for every x ∈ X ′ , then E ( X, f ) is countable.Proof. Let E ∗ = E ( X, f ) \ { f n : n ∈ N } . It suffices to show that E ∗ is countable. First noticethat E ∗ is equal to { f p : p ∈ N ∗ } by Lemma 2.4. Since X ′ is f -invariant (by Lemma 4.1(ii)),then ( X ′ , f ↾ X ′ ) is a well defined dynamical system. By Lemma 4.1(i), E ( X ′ , f ↾ X ′ ) is finite.Let w ∈ X with a dense orbit. Consider the function ϕ : E ∗ → E ( X ′ , f ↾ X ′ ) × X ′ given by ϕ ( g ) = ( g ↾ X ′ , g ( w )). It suffices to show that ϕ is injective. In fact, let g ∈ E ∗ , then g = f p for some p ∈ N ∗ . Notice that every isolated point is of the form f l ( w ) for some l ∈ N . Thus g ( f l ( w )) = f p ( f l ( w )) = f l ( f p ( w )) = f l ( g ( w )). Therefore g is completely determined by g ↾ X ′ and g ( w ). Hence ϕ is injective. (cid:3) Our next example satisfies the second conditions of Theorem 4.2 and all the p -iterates are dis-continuous, for p ∈ N ∗ .The phase space of the next two examples is going to be ω + 1 which, for our convenience, willbe identified with the following subspace of R : X = { d i,j,k : i, j, k ∈ N } ∪ { d j,k : j, k ∈ N } ∪ { d k : k ∈ N } ∪ { d } , where ( d k ) k ∈ N is a strictly increasing sequence such that d k −−−→ k →∞ d ; ( d j, ) j ∈ N is a strictly increasingsequence contained in ( −∞ , d ) such that d j, −−−→ j →∞ d ; for each positive k ∈ N , the sequence( d j,k ) j ∈ N is strictly increasing, it is contained in ( d k − , d k ) and d j,k −−−→ j →∞ d k ; D , := { d i, , : i ∈ N } is a strictly increasing sequence such that d i, , −−−→ i →∞ d , and it is contained in ( −∞ , d , ); D ,k = { d i, ,k : i ∈ N } is a strictly increasing sequence with d i, ,k −−−→ i →∞ d ,k and contained in ( d k − , d ,k )for each k ∈ N \ { } D j,k := { d i,j,k : i ∈ N } is a strictly increasing sequence with d i,j,k −−−→ i →∞ d j,k and contained in ( d j − ,k , d j,k ) for every j ∈ N \ { } and for every k ∈ N .We are ready to describe our first example. Example 4.5.
There is a continuous function f : ω + 1 → ω + 1 such that: (1) O f ( d , , ) is dense. (2) The points d c n − , and d c n +1 , have infinite orbits, where c n = 2 + 3 n for every n ∈ N . (3) | ω f ( x ) | = 1 for every x ∈ ( ω + 1) ′ . (4) E ( ω + 1 , f ) is homeomorphic to ω + 1 . (5) f p is discontinuous for all p ∈ N ∗ .Our function f is defined as follows: a ) f ( d ) = d and f ( d n ) = d n for each n ∈ N . b ) f ( d , ) = d , f ( d , ) = d , , f ( d , ) = d , and f ( d , ) = d . c ) f ( d c n − , ) = d c n − − , for each n > . d ) f ( d c n , ) = d c n − , for each n > . e ) f ( d c n +1 , ) = d c n +1 +1 , for each n ∈ N . f ) f ( d j,k ) = d j − ,k for each j > and k > . g ) f ( d ,k ) = d k − for each k > . h ) f ( d i,j,k ) = d i +1 ,j − ,k for each i ∈ N and j, k > . i ) f ( d i, , ) = d ,i, for each i ∈ N . j ) f ( d i, ,k ) = d ,i +1 ,k − for each i ∈ N and k > . ) f ( d i, , ) = d ,c i +1 − , for each i ∈ N . l ) f ( d i,c , ) = d i +1 , , for each i ∈ N . m ) f ( d i,c n , ) = d i +1 ,c n − , for each i ∈ N and n > . n ) f ( d i,c n − , ) = d i +1 ,c n − − , for each i ∈ N and n > . o ) f ( d , c n +1 ,
0) = d ,c n , for each n ∈ N . p ) f ( d i,c n +1 , ) = d i − ,c n +1 +1 , for each i > and n ∈ N . q ) f ( d i, , ) = d i, , for each i ∈ N . r ) f ( d i, , ) = d i, , for each i ∈ N . s ) f ( d i, , ) = d , ,i +2 for each i ∈ N . t ) f ( d i, , ) = d i +1 , , for each i ∈ N .In the next diagram, we can see the behavior of the orbits on the isolated points. d , , → d , , → d , , → d , , → d , , → d , , → d , , → d , , d , , → d , , → d , , → d , , → d , , → d , , → d , , → d , , → d , , → d , , → d , , → d , , → d , , → d , , d , , → d , , → d , , → d , , → d , , → d , , → d , , → d , , → d , , → d , , → d , , → d , , → d , , → d , , → d , , → d , , → d , , → d , , → d , , → d , , → d , , d , , → d , , → d , , → d , , → d , , → d , , → d , , → d , , → d , , → d , , → d , , → d , , → d , , → d , , → d , , → d , , → d , , → d , , → d , , → d , , → d , , → d , , → d , , → d , , ... ... ... d n, , → d , ,n +2 → d , ,n +1 · · · d , ,n +1 · · · → d n, , → d ,c n +1 − , → · · ·→ d n, , → d n +1 , , · · · ... ... ...The following properties are satisfied. i ) f ( D j,k ) = D j − ,k for each j > and k > . ii ) f ( D c , ) = D , \ { d , , } and f ( D c n , ) = D c n − , \ { d ,c n − , } for each n > . iii ) f ( D c n +1 , ) = D c n +1 +1 , ∪ { d ,c n , } for each n ∈ N . iv ) f ( D c n − , ) = D c n − − , \ { d ,c n − − , } for each n = 0 . v ) f ( D , ) = { d ,c n +1 − , : n ∈ N } . vi ) f ( D , ) = { d , ,k +2 : k ∈ N } . vii ) f ( D , ) = D , and f ( D , ) = D , . viii ) f ( D , ) = { d ,j, : j ∈ N } and f ( D ,k ) = { d ,j +1 ,k − : j ∈ N } for each k > .Observe that clauses a ) and i ) imply that f is continuous at d , at d i for each i > and at d ij for every i, j > . For j > , f is continuous at d j by clauses g ) and viii ) . By d ) , e ) , ii ) , iii ) and iv ) , we have that f is continuous at d and at d i, for every i > . By b ) , v ) , vi ) and vii ) , f iscontinuous at the points d , , d , d and d , . Therefore, f is continuous.It is evident that the orbit O f ( d , , ) is dense. Also it is evident that the points d c n − , and d c n +1 , have infinite orbits, for every n ∈ N . The following relationships follows directly from thedefinition: ) f p ( d ) = d and f p ( d n ) = d n for each n ∈ N and p ∈ N ∗ . II ) f p ( d , ) = d , f p ( d , ) = d and f p ( d , ) = d . III ) f p ( d c n , ) = d for each n > . IV ) f p ( d c n +1 , ) = d = f p ( d c n − , ) for each n ∈ N . V ) f p ( d j,k ) = d k − for each j > and k > .All these properties imply clause (3) , condition (4) follows from Theorem 4.2 and the last condition (5) follows from clauses III ) and IV ) . In the next example, we show that there exists a continuous function f such that the dynamicalsystem ( ω + 1 , f ) is transitive and has a sequence of accumulation points with arbitrarily largeperiod. This example differs from Example 4.1 of [7] since this dynamical system is transitive andin the other one all points have finite orbit. Notice also that, in the example below, all p -iteratesare discontinuous. Example 4.6.
There is a function f : ω + 1 → ω + 1 such that (1) O f ( d , , ) is dense. (2) All points of ( ω + 1) ′ have finite orbit. (3) The accumulation point d a n has period n + 2 , where a = 0 and a n = a n − + n + 1 , for every n ∈ N . (4) E ( ω + 1 , f ) is homeomorphic to N . (5) f p is discontinuous for all p ∈ N ∗ .The function f is defined as follows: a ) f ( d ) = d . b ) f ( d a n ) = d a n + n +1 = d a n +1 − for each n ∈ N . c ) f ( d a n + k ) = d a n + k − for each n ∈ N and < k ≤ n + 1 . d ) f ( d , ) = d and f ( d ,a n ) = d a n − for each n > . e ) f ( d i,a n ) = d i − ,a n + n +1 = d i − ,a n +1 − for each n ∈ N and i > . f ) f ( d i,a n + k ) = d i,a n + k − for each i ∈ N , n ∈ N and < k ≤ n + 1 . g ) f ( d , , ) = d , , and f ( d n, , ) = d , ,a n for each n > . h ) f ( d i,j, ) = d i +1 ,j − , for each i ∈ N and j > . i ) f ( d i, , ) = d i +1 , , for each i ∈ N . j ) f ( d i,j, ) = d i,j, for each i ∈ N and j > . k ) f ( d i,j,a n ) = d i,j − ,a n + n +1 for each i ∈ N and j, n > . l ) f ( d i, ,a n ) = d ,i +1 ,a n − for each i ∈ N and n > . m ) f ( d i,j,k ) = d i,j,k − for each i, j ∈ N and k / ∈ H and k / ∈ { a n + 1 : n ∈ N } . n ) f ( d i,j,k ) = d i,j,k − for each i ∈ N , j > and k ∈ { a n + 1 : n > } . o ) f ( d i, ,k ) = d i +1 , ,k − for each i ∈ N and k ∈ { a n + 1 : n > } .To have some idea about the orbits we describe some of them in the next diagram: d , , → d , , → d , , → d , , → d , , → d , , → d , , → d , , → d , , → d , , → d , , → d , , → d , , → d , , → d , , → d , , → d , , → d , , → d , , → d , , → d , , → d , , → d , , → d , , → d , , → d , , → d , , → d , , → d , , → d , , → d , , → d , , → d , , → d , , → d , , → d , , → d , , → d , , → d , , → d , , → d , , → d , , → d , , → d , , → d , , → d , , → d , , → d , , → d , , → . . . ... ... ... → d n, , → d , ,a n → d , ,a n − · · · d , ,a n − → d , ,a n − · · · → d , ,a n − → d , ,a n − − → · · · → d , ,a n − → d n, , → d n +1 , , → · · · .. ... ...Notice that: i. f ( D , ) = { d , ,a n : n ∈ N } . ii. f ( D j, ) = D j − , \ { d ,j − , } for each j > . ii. f ( D , ) = D , \ { d , , } and f ( D ij ) = D ( i − j for each i, j > . iii. f ( D j, ) = D j, for each j > . iv. f ( D ,a n ) = { d ,j +1 ,a n − : j ∈ N } for each n > . v. f ( D j,a n ) = D j − ,a n + n +1 \ { d ,j − ,a n + n +1 } for each j , n > . vi. f ( D j,k ) = D j,k − for each j ∈ N and k / ∈ H ∪ { a n + 1 : n ∈ N } . vii. f ( D j,k ) = D j,k − for each j > and k ∈ { a n + 1 : n ∈ N } . viii. f ( D ,k ) = D ,k − \ { d , ,k − } for each k ∈ { a n + 1 : n ∈ N } .By conditions a ) , b ) , c ) , iv ) , v ) , vi ) , vii ) and viii ) , we have that f is continuous at d , d k for every k > ; f is continuous at d by conditions b ) and ii ) ; and f is continuous at d ij ,by conditions d ) , e ) , f ) and identities from i ) to viii ) for each i, j ∈ N . Consider the sequence ( d i ) i ∈ N that convergesto d . By conditions d ) , e ) and f ) , for every i ∈ N there exist l i ∈ N such that f l i ( d i ) = d . Then,for each i ∈ N , p ∈ N ∗ f p ( d i ) = d and f p ( d ) = d . So, we conclude that f p is discontinuous at d for every p ∈ N ∗ . Since there are periodic points of arbitrarily large period, the Ellis semigroup E ( ω + 1 , f ) is homeomorphic to N (by Theorem 2.7 of [8] ). Concerning the previous example we formulate the following question.
Question 4.7.
Is there a continuous function f : ω + 1 → ω + 1 such ( ω + 1 , f ) is transitiveand E ( ω + 1 , f ) ∗ contains both continuous and discontinuous functions? With respect to Theorem 4.2, it is natural to ask the following.
Question 4.8.
Let ( ω α +1 , f ) be a transitive dynamical systems where α ≥ is a countable ordinal.If < sup {| ω f ( x ) | : x ∈ ( ω α + 1) ′ } < ω , is E ( ω α + 1 , f ) homeomorphic to ω β + 1 for some countableordinal β ≥ ? References
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E-mail address : [email protected] Escuela de Matem´aticas, Facultad de Ciencias, Universidad Industrial de Santander, Ciudad Uni-versitaria, Carrera 27 Calle 9, Bucaramanga, Santander, A.A. 678, Colombia and Departamento deMatem´aticas, Facultad de Ciencias, Universidad de los Andes, M´erida 5101, Venezuela
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