The ideal boundary and the accumulation lemma

TL;DR

This paper introduces the accumulation lemma, establishing that hyperbolic branches of periodic points are contained within certain compact sets on surfaces, supported by a classification of surfaces and residual domain characterizations.

Contribution

It develops a classification of connected surfaces with boundary and characterizes residual domains, enabling the proof of the accumulation lemma for measure-preserving homeomorphisms.

Findings

Proves the accumulation lemma for hyperbolic periodic points.

Classifies connected surfaces with boundary.

Characterizes residual domains of compact subsets.

Abstract

Let $S$ be a connected surface possibly with boundary, $\mu$ a finite Borel measure which is positive on open sets and $f:S\to S$ a homeomorphism preserving $\mu$. We prove that if $K$ is a compact connected subset of $S$ and $L$ is a branch of a hyperbolic periodic point if $f$ then $L\cap K\ne\emptyset$ implies $L\subset K$. This is called the accumulation lemma. For this we develop a classification of connected surfaces with boundary and a characterization of residual domains of compact subsets with finitely many connected components in a connected surface with boundary.