Birkhoff spectrum for piecewise monotone interval maps

TL;DR

This paper studies the Birkhoff spectra for regular potentials in piecewise monotone interval maps, providing complete results for hyperbolic cases and partial descriptions for parabolic cases, extending Hofbauer's work.

Contribution

It extends Hofbauer's work by analyzing Birkhoff spectra in non-uniformly hyperbolic interval maps, including parabolic behavior.

Findings

Complete Birkhoff spectrum results for hyperbolic maps

Partial description of spectra with positive Lyapunov exponent in parabolic cases

Lower bounds on the full spectrum in non-uniform cases

Abstract

For piecewise monotone interval maps we look at Birkhoff spectra for regular potential functions. This means considering the Hausdorff dimension of the set of points for which the Birkhoff average of the potential takes a fixed value. In the uniformly hyperbolic case we obtain complete results, in the case with parabolic behaviour we are able to describe the part of the sets where the lower Lyapunov exponent is positive. In addition we give some lower bounds on the full spectrum in this case. This is an extension of work of Hofbauer on the entropy and Lyapunov spectra.