An interval map with a spectral gap on Lipschitz functions, but not on bounded variation functions

TL;DR

This paper constructs a specific interval map that demonstrates a spectral gap on Lipschitz functions but not on bounded variation functions, highlighting differences in spectral properties across function spaces.

Contribution

It provides a novel example of an interval map with contrasting spectral gap properties on Lipschitz and bounded variation spaces.

Findings

Spectral gap exists on Lipschitz functions

No spectral gap on bounded variation functions

Map preserves Lebesgue measure

Abstract

We construct a uniformly expanding map of the interval, preserving Lebesgue measure, such that the corresponding transfer operator admits a spectral gap on the space of Lipschitz functions, but does not act continuously on the space of bounded variation functions.