Topological conditions for the existence of absorbing Cantor sets

Bruin, Henk

doi:10.1090/S0002-9947-98-02109-6

Topological conditions for the existence of absorbing Cantor sets
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Trans. Amer. Math. Soc. 350 (1998), 2229-2263 Request permission

Abstract:

This paper deals with strange attractors of S-unimodal maps $f$. It generalizes earlier results in the sense that very general topological conditions are given that either

[i)] guarantee the existence of an absorbing Cantor set provided the critical point of $f$ is sufficiently degenerate, or
[ii)] prohibit the existence of an absorbing Cantor set altogether.

As a by-product we obtain very weak topological conditions that imply the existence of an absolutely continuous invariant probability measure for $f$.

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