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