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Factor maps and invariant distributional chaos. (English) Zbl 1345.37006

This article presents two main results. The first one confirms previous belief that the specification property might be a sufficient condition for the existence of distributionally scrambled sets:
Theorem 16. Let \(X\) be a compact metric space that is not a singleton. Let \(f: X\rightarrow X\) be a continuous surjection with the specification property and a fixed point \(p\). Then there exists \(\varepsilon>0\) and a dense Mycielski distributionally \(\varepsilon\)-scrambled set \(D\) such that \(f(D)\subset D\) and \(p\in D\). Additionally, the set \(D\) is contained in the union of recurrent points of \(f\), points with dense orbits in \(X\), and \(\{p\}\).
The second result shows that a semiconjugacy which covers a fixed point finite-to-one transfers invariant distributionally scrambled sets:
Theorem 17. Let \(Y\) be an infinite mixing sofic shift with a fixed point \(p\in Y\) and let \(\pi: (X,f)\rightarrow (Y,\sigma)\) be a factor map such that \(\pi^{-1}(p)\) is finite and consists of periodic points. Then for some \(\varepsilon>0\) there is a Cantor distributionally \(\varepsilon\)-scrambled set \(C\) for \(f\). If additionally \(\pi^{-1}(p)\) consists of fixed points then there is an invariant Mycielski distributionally \(\varepsilon\)-scrambled set \(M\) for \(f\).
The paper concludes with a practical application of the developed theory. It is proven that the Poincaré map induced by the time-periodic local process generated by the nonautonomous ODE \[ \dot{z}=(1+e^{i\kappa t}|z|^2)\overline{z}^3-N \] is chaotic (here \(N\) and \(\kappa\) are small positive real numbers).

MSC:

37B05 Dynamical systems involving transformations and group actions with special properties (minimality, distality, proximality, expansivity, etc.)
37B10 Symbolic dynamics
37C25 Fixed points and periodic points of dynamical systems; fixed-point index theory; local dynamics
37F20 Combinatorics and topology in relation with holomorphic dynamical systems
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