Convergence of the Birkhoff spectrum for nonintegrable observables

TL;DR

This paper investigates the Birkhoff spectrum for interval maps with countably many branches and polynomial tail observables, showing its real analyticity and dependence on tail exponents, extending prior regularity results.

Contribution

It extends previous work by analyzing the Birkhoff spectrum's convergence and analyticity for less regular observables with polynomial tails, using thermodynamic formalism.

Findings

Birkhoff spectrum is real analytic for the considered maps.

Convergence to Hausdorff dimension is governed by tail exponent.

Applications include Gauss maps, L"uroth transformations, and induced Manneville-Pomeau maps.

Abstract

We consider interval maps with countably many full branches and observables with polynomial tails. We show that the Birkhoff spectrum is real analytic and that its convergence to the Hausdorff dimension of the repeller is governed by the polynomial tail exponent. This result extends previous work by Arima on more regular observables and demonstrates how the tail behaviour influences the structure of the Birkhoff spectrum. Our proof relies on techniques from thermodynamic formalism and tail estimates for the observable and our applications are to natural observations on Gauss maps, L\"uroth transformations as well as to a the first return time for a class of induced Manneville-Pomeau maps.