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Dynamical system topology preserved in the presence of noise. (English) Zbl 0956.37012

The paper generalizes Smale horseshoe dynamics. The definitions of “topological horseshoe” – \(F\) and “generalized quadrilateral” – \(Q\) are given. The behavior of restriction \(F\) on \(Q\) of a locally compact, separable, locally connected metric space is examined. The permanent set consists of points whose entire trajectory is in \(Q\), the entrainment set consists of points whose backward trajectory is in \(Q\). Theorem 3 describes the dynamics of \(F\) on the permanent set. In particular, the permanent set is a quotient Cantor set on which the restriction \(F\) factors over the shift on \(M\) symbols. Theorem 4 describes the analogous dynamics under additional noise when iteration \(F^n\) is replaced by its perturbation \(F_n\). Theorem 5 describes conditions when the entrainment set contains the invariant indecomposable continuum. Theorem 6 describes the analogous result under additional noise.

MSC:

37B99 Topological dynamics
37H99 Random dynamical systems
54H20 Topological dynamics (MSC2010)
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