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Dynamics forced by surface trellises

Pieter Collins
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Abstract

Given a saddle fixed point of a surface diffeomorphism, its stable and unstable curves $W^S$ and $W^U$ often form a homoclinic tangle. Given such a tangle, we use topological methods to find periodic points of the diffeomorphism, using only a subset of the tangle with finitely many points of intersection, which we call a trellis. We typically obtain exponential growth of periodic orbits, symbolic dynamics and a strictly positive lower bound for topological entropy. For a simple example occurring in the Henon family, we show that the topological entropy is at least 0.527.

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Pieter Collins. 1999-07-13. Dynamics forced by surface trellises. https://arxiv.org/abs/math/9907086

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