Flattening Functions on Flowers

TL;DR

This paper studies Lipschitz functions on the circle and characterizes when they can be simplified to constants on certain structured sets called flowers, revealing the geometric structure of these functions and their relation to maximizing measures.

Contribution

It introduces the concept of flattening functions on flowers, characterizes the space of functions that can be flattened, and links this to Sturmian measures and their associated flowers.

Findings

The space of flattenable functions on a p-flower has codimension p.

Explicit linear constraints for flattening functions are derived.

Functions with Sturmian maximizing measures can be flattened on a 1-flower.

Abstract

Let $T$ be an orientation-preserving Lipschitz expanding map of the circle $\T$. A pre-image selector is a map $\tau:\T\to\T$ with finitely many discontinuities, each of which is a jump discontinuity, and such that $\tau(x)\in T^{-1}(x)$ for all $x\in\T$. The closure of the image of a pre-image selector is called a flower, and a flower with $p$ connected components is called a $p$-flower. We say that a real-valued Lipschitz function can be Lipschitz flattened on a flower whenever it is Lipschitz cohomologous to a constant on that flower. The space of Lipschitz functions which can be flattened on a given $p$-flower is shown to be of codimension $p$ in the space of all Lipschitz functions, and the linear constraints determining this subspace are derived explicitly. If a Lipschitz function $f$ has a maximizing measure $S$ which is Sturmian (i.e. is carried by a 1-flower), it is shown that $f$ can be Lipschitz flattened on some 1-flower carrying $S$.