Dimension of self-conformal measures associated to an exponentially separated analytic IFS on $\mathbb{R}$

Abstract

We extend Hochman's work on exponentially separated self-similar measures on $\mathbb{R}$ to the real analytic setting. More precisely, let $Φ=\left\{ φ_{i}\right\} _{i\inΛ}$ be an iterated function system on $I:=[0,1]$ consisting of real analytic contractions, let $p=(p_{i})_{i\inΛ}$ be a positive probability vector, and let $μ$ be the associated self-conformal measure. Suppose that the maps in $Φ$ do not have a common fixed point, $0<\left|φ_{i}'(x)\right|<1$ for $i\inΛ$ and $x\in I$, and $Φ$ is exponentially separated. Under these assumptions, we prove that $\dimμ=\min\left\{ 1,H(p)/χ\right\} $, where $H(p)$ is the entropy of $p$ and $χ$ is the Lyapunov exponent. The main novelty of our work lies in an argument that reduces convolutions of $μ$ with measures on the (infinite-dimensional) space of real analytic maps to convolutions with measures on vector spaces of polynomials of bounded degree. The reason for this reduction is that, for the latter convolutions, we can establish an entropy increase result, which plays a crucial role in the proof. We believe that our proof strategy has the potential to extend other significant recent results in the dimension theory of stationary fractal measures to the real analytic setting.