Dynamics forced by surface trellises

1999/07/13 by Pieter Collins, Collins, Pieter
Engineering · Mathematics · #58F15 #Dynamical Systems (math.DS) #FOS: Mathematics #Fluid Dynamics Simulations and Interactions #Granular flow and fluidized beds #Vibration and Dynamic Analysis #math.DS #msc:58F15

paper · pdf · doi:10.48550/arxiv.math/9907086

22 pages, 15 figures. Confeerence proceedings: Topology in Dynamics, Wake Forest University, Winston-Salem, 1998

arxiv created 1999/07/13 · openalex publication_date 1999/07/13 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

Related

• Dynamics of surface diffeomorphisms relative to homoclinic and heteroclinic orbits
• Dynamics of surface homeomorphisms Topological versions of the Leau-Fatou flower theorem and the stable manifold theorem
• Hofer's geometry and topological entropy
• Topological horseshoes for surface homeomorphisms
• Symmetric homoclinic tangles in reversible dynamical systems have positive topological entropy