Entropy of convolutions on the circle

Abstract

Given ergodic $p$-invariant measures $\{\mu_i\}$ on the $1$-torus $\mathbb{T} = \mathbb{R}/\mathbb{Z}$, we give a sharp condition on their entropies, guaranteeing that the entropy of the convolution $\mu_1\ast\cdots\ast \mu_n$ converges to $\log p$. We also prove a variant of this result for joinings of full entropy on $\mathbb{T}^{\mathbb{N}}$. In conjunction with a method of Host, this yields the following. Denote $\sigma_1(x) = qx$ (mod $1$). Then for every $p$-invariant ergodic $\mu$ with positive entropy, $\frac{1}{N} \Sigma_{n=0}^{N-1}\sigma_{c_n}\mu$ converges weak${}^\ast$ to Lebesgue measure as $N\to \infty$, under a certain mild combinatorial condition on $\{c_k\}$. (For instance, the condition is satisfied if $p = 10$ and $c_k = 2^k+6^k$ or $c_k = 2^{2^k}$.) This extends a result of Johnson and Rudolph, who considered the sequence $c_k = q^k$ when $p$ and $q$ are multiplicatively independent.

We also obtain the following corollary concerning Hausdorff dimension of sum sets: For any sequence $\{S_i\}$ of $p$-invariant closed subsets of $\mathbb{T}$, if $\Sigma \dim _H (S_i)/|\log \dim _H(S_i)| =\infty$, then $\dim_H(S_1+\cdots + S_n) \to 1$.