STATISTICAL CONVERGENT TOPOLOGICAL SEQUENCE ENTROPY MAPS OF THE CIRCLE

A continuous map f of the interval is chaotic iff there is an increasing of nonnegative integers T such that the topological sequence entropy of f relative to T, hT(f), is positive [4]. On the other hand, for any increasing sequence of nonnegative integers T there is a chaotic map f of the interval such that hT(f)=0 [7]. We prove that the same results hold for maps of the circle. We also prove some preliminary results concerning statistical convergent topological sequence entropy for maps of general compact metric spaces.


INTRODUCTION
Let (X, ρ ) be a compact metric space; denote by the space of all maps of this space into itself.We will pay a special attention to the case when X is the circle .By N we denote the set of all positive integers.If is an arbitrary sequence of nonnegative integers then the (T,f,n)-trajectory of x∈ X is the sequence . The set of all periodic points of f is denoted by Per(f) and the set of periods of all periodic points of f by P(f).A set A ⊆ X is called a retract of X if there is a map r : X → A such that r(a) = a for every a∈A.
Definition1: Let (X, ρ ) be a compact metric space.The ∞ =1 , if for 0 > ε , and for Then st-Sep(T,f) = st-Span(T,f) we define the statistical convergent topological sequence entropy of f relative to T, h st-T (f), to be st-Sep(T, f) [3].
In [4] Franzová and Smítal, proved that a map f of the interval is chaotic if and only if there is an increasing sequence of nonnegative integers T such that h T (f) > 0. A natural question arose whether there is some universal sequence which characterizes chaos.This is not the case as it was proved in [7] for any increasing sequence of nonnegative integers T there is a chaotic map f with h T (f) = 0.The main aim of this paper is to prove the same results for statistical convergent topological sequence entropy maps of the circle.
is chaotic if and only if there is an statistical convergent sequence of nonnegative integers T such that h st-T (f) > 0. Theorem2: Let X be a compact metric space containing a homeomorphic image of an interval and let T be an statistical convergent sequence of nonnegative integers.Then there is a chaotic map

PRELIMINARY RESULTS
Let (X, ρ ) and (Y,σ ) be compact metric spaces, In this situation we have the following.
Lemma1: Let T be an increasing sequence of nonnegative integers.
(i).We have that π is a homeomorphism betweenX and .X π Thus, by (iii), , T be an statistical convergent sequence of nonnegative integers and k be a positive integer.Then there is an statistical convergent statistical sequence of nonnegative integers S such that are equicontinuous, i. e., for any ε > 0 there is δ =δ ( ε )>0 such that if x,y∈X and ρ (x; y) δ ≤ then We may assume that .
and k be a positive integer.Then the following two conditions are equivalent: (i) there is an increasing sequence T of nonnegative integers such that h T (f) >0; (ii) there is an increasing sequence T of nonnegative integers such that h T (f k ) >0.
In the sequel we will discuss the space of maps of the circle.The space ) (S C can be decomposed into the following classes [1], [10] According to this we will distinguish three different cases.

MAPS WITHOUT PERIODIC POINTS
Throughout this section we assume ) (S C f ∈ to have no periodic point.We are going to show that Theorem1 holds for such continuous maps.Since, by [9], f is not chaotic, we need only to show that 0 . Otherwise, there is a nowhere dense perfect set E which is the only ω -limit set of f, all (closed) contiguous intervals are wandering, the preimage of any contiguous interval is a contiguous interval, the image of any contiguous interval is either a contiguous interval or a point from E. Moreover, f| E is monotone.By linear extension of f| E we obtain a monotone map ) (S C g ∈ . By [8], h T (g)=0 By Lemma1(i),

Entropy2004, 6
For E x ∉ put y to be an endpoint of the contiguous interval which contains x.We are going to show that there is a set of cardinality at most n.k.N k which ) , , , ( n f T ε st-spans all considered points.It is sufficient to show that there is a set with cardinality at most Each element in this trajectory is either a contiguous interval or a point from E. At most k of them have lengths greater than or equal to ε -cut each of such elements to N segments shorter thanε .All the other elements of the trajectory will be considered to be segments themselves.To each x∈I(t i I j ) assign its code-the sequence (S 1 (x),…, S n (x)) where ) (x S l is the segment containing l t f x.We have at most N k different codes and all points with the same code can be (T, f, ε , n)-st-spanned by one point.From what has been said above we see that

MAPS WITH PERIODIC POINTS
We will first deal with the case C 2 .We know that for any n∈N f is chaotic if and only if n f is chaotic.Taking into account Corollary1 we can without loss of generality assume that ,...}.,2 {1,2, Since f has a fixed point, by [10] there is a lifting F and an F-invariant compact interval J longer than 1.In the following discussion of the case C 2 we will write F and Π instead of
Lemma3: Let F be chaotic.Then there is an statistical convergent sequence T such that where k>1 is odd then, by Sharkovsky theorem, it has also a periodic point of period So F is of type ∞ 2 , chaotic.By [10] there is an orbit of periodic intervals of period p>diam J such that F p is chaotic on each of them.At least one interval K in this orbit is shorter than 1.Then Π | K is injective and so There is an statistical convergent sequence of nonnegative integers S such that h st-S ( Proof of Theorem1:We are going to show that Theorem 1 holds for maps from the class C 2 .Let f∈C 2 be chaotic.Then we obtain the required result using Lemma 1 and Lemma2.Now let f∈C 2 and let there be an statistical convergent sequence of nonnegative integers T such that h st-T (f)>0.Lemma(ii) implies that h st-T (F)>0 where F has the same meaning as above.F is chaotic.Finally we will discuss the situation for maps from the remaining class C 3 .So let P st-( n f )=N for some n.By [2] We have that h st-( n f ) is positive and so is h st-(f).Then we have that n m f . is strictly turbulent for a suitable N m ∈ which implies that f is chaotic for the same reason as in the interval case.This finishes the proof of Theorem1.

Proof of Theorem2:
The space X contains a homeomorphic image J of the interval [0,1].The set J is a retract of X by [6].Let r : X→J be a corresponding retraction.By [7] there is a chaotic onto map g∈C([0,1]) such that h st-T (g) = 0. Let g ~∈C(J) be a map topologically conjugate with g.Define f∈C(X) by , we have that h st-T (f) = h st-T (f| J ) = 0.
the natural projection of the real line R onto S, i.e., fix an arbitrary ε > 0. We are going to estimate st-Span )contiguous intervals longer than .

2 ε=
Let .Take any point x whose (T,f, n)-trajectory lies in S \ .If x∈E then x is (T, f, ε , n)st-spanned by A.

2 εFix
by a point z∈A, the set A obviously ) points x.So it remains to consider those points whose (T, f, N∈N such that N > . 1 ε proof of Theorem1 for maps without periodic points.

π
F| J and Π | J , respectively, as in the next commutative diagram π Note that if x,y∈J then || Π x, Π y|| ≤ |x-y| with the equality whenever |x-y| .2 1 .