Thermodynamic formalism and multifractal analysis of Birkhoff averages for non-uniformly expanding interval maps with finitely many branches

TL;DR

This work analyzes the multifractal spectrum of Birkhoff averages for non-uniformly expanding interval maps with finitely many branches and parabolic fixed points. It develops a refined thermodynamic formalism by passing to an admissible induced map $\tilde{f}$ and a countable Markov coding, enabling real-analytic dependence of the induced pressure $p(b,q)$ on $(b,q)$ and the existence/uniqueness of equilibrium measures for $-q\phi-b\log|f'|$ with $\lambda(\mu)>0$. The main results describe the Birkhoff spectrum $b(\alpha)$: it is real-analytic and strictly monotone on $(\alpha_{\min},\alpha_{\max})\setminus A$, with $b(\alpha)=\delta$ on the exceptional set $A$, and the spectrum is realized by unique ergodic measures via a conditional variational principle; a Bowen-type formula links $\delta$ to the pressure $P(-\delta\log|f'|)$. The analysis generalizes previous single-parabolic results to multiple parabolic fixed points and provides a framework to obtain the full spectrum together with the associated inducing-time/return-time structure.

Abstract

In this paper, we perform a multifractal analysis of Birkhoff averages for interval maps with finitely many branches and parabolic fixed points. Using the thermodynamic approach, we strengthen the results of Johansson et al. on the conditional variational principle for the multifractal spectra of Birkhoff averages. To do this, we develop several refined properties of the thermodynamic formalism for non-uniformly expanding interval maps.

Theorems & Definitions (40)

• Example 1.1
• Example 1.2
• Theorem 1.3
• Remark 2.1
• Theorem 2.2
• Proposition 2.3
• Theorem 2.4
• Theorem 2.5
• Theorem 2.6
• Lemma 2.7