Multifractal analysis of Lyapunov exponent for the backward continued fraction map

TL;DR

This paper investigates the multifractal spectrum of Lyapunov exponents for interval maps with infinitely many branches, revealing unbounded domains, potential non-differentiability, and non-concavity, contrasting with hyperbolic cases.

Contribution

It provides new insights into the multifractal spectrum and thermodynamic formalism for non-hyperbolic interval maps with parabolic fixed points.

Findings

Spectrum domain is unbounded unlike hyperbolic cases

Points of non-differentiability may exist in the spectrum

Pressure function is real analytic in a certain interval and then zero

Abstract

In this note we study the multifractal spectrum of Lyapunov exponents for interval maps with infinitely many branches and a parabolic fixed point. It turns out that, in strong contrast with the hyperbolic case, the domain of the spectrum is unbounded and points of non-differentiability might exist. Moreover, the spectrum is not concave. We establish conditions that ensure the existence of inflection points. We also study the thermodynamic formalism for such maps. We prove that the pressure function is real analytic in a certain interval and then it becomes equal to zero.