When is a transitive map chaotic?

*(English)*Zbl 0861.54034
Bergelson, V. (ed.) et al., Convergence in ergodic theory and probability. Papers from the conference, Ohio State University, Columbus, OH, USA, June 23–26, 1993. Berlin: de Gruyter. Ohio State Univ. Math. Res. Inst. Publ. 5, 25-40 (1996).

Relations among various properties of topological discrete-time dynamical systems are considered. A dynamical system is a pair \((X,f)\) consisting of a topological space \(X\) and a continuous map \(f:X \to X\). It is assumed in the paper that \(X\) is compact and metrizable. The dynamical system \((X,f)\) is called topologically transitive if for some point \(x\in X\) the omega-limit set \(\omega (x)\) coincides with \(X\). Such a point \(x\) is called a transitive point. \((X,f)\) is called minimal if every point of \(X\) is a transitive point. A point \(x\in X\) is called a minimal point if \(f\) restricted to the closure of the trajectory of \(x\) is a minimal system. \((X,f) \) is called equicontinuous if the family of maps \(\{f^n\}_{n \in\mathbb{N}}\) is uniformly equicontinuous. A point \(x\in X\) is called an equicontinuity point if the family \(\{f^n\}\) is equicontinuous at \(x\). \((X,f)\) is called almost equicontinuous if it has at least one equicontinuity point. \((X,f)\) is called sensitive if there exists an \(\varepsilon>0\) such that for any nonempty open set \(U \subset X\) there exist \(x,y\in U\) such that for some positive integer \(n\) the distance between \(f^n(x)\) and \(f^n(y)\) is greater then \(\varepsilon\). \((X,f)\) is called uniformly rigid if \(\{f^{n_k}\}\) converges uniformly to the identity of \(X\), for some sequence \(n_k\to\infty\).

Among others, the following results are proved in the paper:

Theorem 1. If a dynamical system is topologically transitive but not minimal then the set of nontransitive points is dense.

Theorem 2. If a dynamical system is almost equicontinuous then the set of equicontinuity points coincides with the set of transitive points. In particular, a minimal almost equicontinuous system is equicontinuous.

Theorem 3. If a dynamical system has no equicontinuity points then it is sensitive. In particular, a minimal system is either equicontinuous or sensitive.

Theorem 4. Let a dynamical system be topologically transitive but not minimal. If the set of its minimal points is dense then the system is sensitive.

(Sensitiveness is a kind of chaotic behaviour, hence Theorem 4 answers to the title of the paper.)

Theorem 5. If the dynamical system \((X,f)\) is almost equicontinuous then \(f\) is a homeomorphism, \((X,f)\) is uniformly rigid, every point of \(X\) is a transitive point if and only if it is an equicontinuity point, and the set of transitive points is residual.

An example of nonminimal almost equicontinuous system is given.

For the entire collection see [Zbl 0846.00030].

Among others, the following results are proved in the paper:

Theorem 1. If a dynamical system is topologically transitive but not minimal then the set of nontransitive points is dense.

Theorem 2. If a dynamical system is almost equicontinuous then the set of equicontinuity points coincides with the set of transitive points. In particular, a minimal almost equicontinuous system is equicontinuous.

Theorem 3. If a dynamical system has no equicontinuity points then it is sensitive. In particular, a minimal system is either equicontinuous or sensitive.

Theorem 4. Let a dynamical system be topologically transitive but not minimal. If the set of its minimal points is dense then the system is sensitive.

(Sensitiveness is a kind of chaotic behaviour, hence Theorem 4 answers to the title of the paper.)

Theorem 5. If the dynamical system \((X,f)\) is almost equicontinuous then \(f\) is a homeomorphism, \((X,f)\) is uniformly rigid, every point of \(X\) is a transitive point if and only if it is an equicontinuity point, and the set of transitive points is residual.

An example of nonminimal almost equicontinuous system is given.

For the entire collection see [Zbl 0846.00030].

Reviewer: R.Srzednicki (Kraków)