Nonautonomous dynamical systems

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It seems more likely that for different purposes different analogues of ergodicity of varying strength will fit. Another interesting question is under which conditions there exist reasonably small generating sets for the Misiurewicz class. It is an interesting and probably very far-reaching question to which extent such results can be transferred to the nonautonomous case.

It might be an interesting topic for future research to look for generalizations of the known results about metric sequence entropy. My gratitude is to Tomasz Downarowicz who pointed out Example 18 to me, and to the anonymous referee who made several good suggestions which helped me to improve the paper and who also brought some interesting literature to my attention. Adler, A. Konheim, M. McAndrew, Topological entropy.

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Coherent sets for nonautonomous dynamical systems

Kolyada, L. Snoha, Topological entropy of nonautonomous dynamical systems. Dynamics 4 , no. Kolyada, M. Misiurewicz, L. Snoha, Topological entropy of nonautonomous piecewise monotone dynamical systems on the interval. Krzyzewski, W. Szlenk, On invariant measures for expanding differentiable mappings. Studia Math. Kushnirenko, On metric invariants of entropy type.

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Theory Appl. Zhu, Z. Liu, X. Xu and W. Zhang, Entropy of nonautonomous dynamical systems. Korean Math. Zhu, J.

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He, Topological entropy of a sequence of monotone maps on circles. DOI: Introduction In the theory of dynamical systems, entropy Is an Invariant which measures the exponential complexity of the orbit structure of a system. Preliminaries 2. U1 We leave the easy proof that this number coincides with the topological entropy as defined above to the reader. Remark 1. Proposition 2. Lemma 3. It follows that ,. To obtain the last equality we used Lemma 4.

Since this holds for every U1l 6 C, X1l , the desired inequality follows. Here are three examples: i Topological entropy for uniformly continuous maps on noncompact metric spaces cf. Remark 7. QeQ PeV Some well-known properties of the conditional entropy are summarized in the following proposition cf.

Proposition 8. Let V, Q and K be partitions of X. Proposition 9. Remark Corollary Example Definition Proposition Invariance and Restrictions In order to be a reasonable quantity, the metric entropy of a system f1sxi should be an Invariant with respect to Isomorphlms.

Now assume that is a sequence of partitions for Y1 which is coarser than Q1.

The Misiurewicz Class In this subsection, we Introduce a special admissible class which we will use to prove the variational Inequality. If f1iXI is equicontinuous, then EM is an admissible class. Define a Hn Pn. I, It is clear that Dn i is a compact subset of Qn i. Jm-1 X Dr, l The Variational Inequality Now we are in position to prove the general variational inequality following the lines of Misiurewicz's proof [19].

Theorem Large Misiurewicz Classes Up to now, we only know that the Misiurewicz class contains the trivial sequence of partitions. JN,J1 ,J Lemma Concluding Remarks and Open Questions In this paper, we introduced a notion of metric entropy for quite general nonautonomous dynamical systems and studied its elementary properties, in particular its relation to the topological entropy defined by Kolyada, Misiurewicz, and Snoha.

Acknowledgements My gratitude is to Tomasz Downarowicz who pointed out Example 18 to me, and to the anonymous referee who made several good suggestions which helped me to improve the paper and who also brought some interesting literature to my attention. References [1] R. Nonlinearity 24 , no. User agreement Privacy policy.

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