Conditions implying annular chaos

TL;DR

This paper establishes new verifiable conditions for topological chaos in annular homeomorphisms, enabling computer-assisted proofs and advancing understanding of the relationship between entropy and rotation sets.

Contribution

It introduces a novel set of sufficient conditions based on rotation numbers and invariant continua, facilitating practical verification of chaos.

Findings

Conditions enable computer-assisted chaos proofs

Applied to well-known examples and analytic families

Supports conjecture linking entropy and rotation sets

Abstract

This work investigates topological chaos for homeomorphisms of the open annulus, introducing a new set of sufficient conditions based on points with distinct rotation numbers and their topological relation to invariant continua. These conditions allow us to formulate classic methods for verifying annular chaos in a finitely verifiable version supported on basic properties of the map. The results pave the way for simple computer-assisted proofs of chaos in a wide range of annular maps, including many well known examples, and we present these proofs for some analytic families, demonstrating the effectiveness of the method. On the theoretical side, one of the consequences of the established conditions permits the proof of a folkloric conjecture about the relation between topological entropy and rotation sets.