Spectral gap property for random dynamics on the real line and multifractal analysis of generalised Takagi functions

TL;DR

This paper studies the spectral gap property for random expanding diffeomorphisms on the real line and applies it to perform a detailed multifractal analysis of generalized Takagi functions.

Contribution

It establishes the spectral gap property for the transition operator of random dynamics and introduces a multifractal analysis of new generalized Takagi functions.

Findings

Spectral gap property proven for the transition operator.

Complete multifractal spectrum of generalized Takagi functions obtained.

New insights into the regularity and fractal structure of these functions.

Abstract

We consider the random iteration of finitely many expanding $\mathcal{C}^{1+\epsilon}$ diffeomorphisms on the real line without a common fixed point. We derive the spectral gap property of the associated transition operator acting on H\"older spaces. As an application we introduce generalised Takagi functions on the real line and we perform a complete multifractal analysis of the pointwise H\"older exponents of these functions.