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

TL;DR

This work extends Hochman’s dimension theory to real-analytic IFS on $\mathbb{R}$ under exponential separation. The authors prove that the self-conformal measure $\mu$ attached to the analytic IFS and a positive probability vector $p$ satisfies $\dim\mu=\min\{1, H(p)/\chi(\Phi,p)\}$, with the analyticity assumption being essential for their Taylor-polynomial reduction and entropy analysis. The core innovation is reducing convolutions with measures on the infinite-dimensional analytic map space to convolutions with measures on finite-dimensional polynomial spaces, where an entropy-increase mechanism can be applied via Hochman’s inverse theorem. The paper also derives consequences for the dimension of self-conformal sets and for one-parameter analytic families, and outlines potential extensions of these techniques to broader results in stationary fractal measures. This provides a robust framework for understanding fractal dimensions in the real-analytic setting, with implications for exact overlaps and parametric stability.

Abstract

We extend Hochman's work on exponentially separated self-similar measures on $\mathbb{R}$ to the real analytic setting. More precisely, let $Φ=\left\{ \varphi_{i}\right\} _{i\inΛ}$ 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, $0<\left|\varphi_{i}'(x)\right|<1$ 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.

Theorems & Definitions (30)

• Definition 1.1
• Theorem 1.2
• Corollary 1.3
• proof
• Corollary 1.4
• Remark
• proof
• Lemma 2.1
• Remark
• proof