Expanding Maps on Flowers, Interval Exchange Transformations, and Ergodic Optimization

TL;DR

This paper explores expanding maps on flowers, their relation to interval exchange transformations, and their role in ergodic optimization, including numerical evidence for trigonometric polynomials maximizing measures on flowers.

Contribution

It extends the understanding of flowers in ergodic optimization and links them to interval exchange transformations, providing new insights and numerical support.

Findings

Sets in flowers have at most linear complexity.

Flowers relate to a special class of interval exchange transformations.

Numerical results support trigonometric polynomials being maximized on flowers.

Abstract

In this paper, we discuss expanding maps on a class of invariant sets called flowers. We show that any set contained in a flower has at most linear complexity, and we present a relationship between flowers and a special class of interval exchange transformations. This extends work of Bullett and Sentenac, who showed that any Sturmian system may be embedded into the circle as a doubling-invariant subset that is contained in a half circle. Flowers were first introduced in the context of ergodic optimization, as candidate sets for supporting maximizing measures. We discuss the relationship to ergodic optimization, and present numerical results that support the conjecture that trigonometric polynomials are maximized on flowers.