On the dynamics of Translated Cone Exchange Transformations

TL;DR

This paper studies translated cone exchange transformations, revealing their complex dynamics including invariant sets, periodic islands, and non-ergodic behavior near the origin, by analyzing their first return maps and embeddings of interval exchange transformations.

Contribution

It introduces and analyzes a new family of piecewise isometries, establishing the existence of invariant sets and periodic islands, and linking embeddings of interval exchange transformations to complex dynamics.

Findings

Existence of infinitely many bounded invariant sets.

Presence of infinitely many periodic islands.

Non-ergodic behavior near the origin.

Abstract

In this paper we investigate translated cone exchange transformations, a new family of piecewise isometries and renormalize its first return map to a subset of its partition. As a consequence we show that the existence of an embedding of an interval exchange transformation into a map of this family implies the existence of infinitely many bounded invariant sets. We also prove the existence of infinitely many periodic islands, accumulating on the real line, as well as non-ergodicity of our family of maps close to the origin.