Entropy of convolutions on the circle

Elon Lindenstrauss, David Meiri and Yuval Peres

Review from Zentralblatt MATH:

This paper investigates the entropy for convolutions of $p$-invariant measures on the circle and their ergodic components. In particular the following two theorems are proved:

Theorem 1: Let $\{\mu_i\}$ be a countably infinite sequence of $p$-invariant ergodic measures on the circle whose normalized base-$p$ measures, $h_i= h(\mu_i,\sigma_p)/\log p$, satisfy $\sum h_i/|\log h_i|= \infty$. Then $h(\mu_1*\cdots* \mu_n,\sigma_p)$ tends to $\log p$ monotonically as $n$ tends to $\infty$. In particular $\mu_1*\cdots* \mu_n$ tends to $\lambda$ weak$^*$.

Theorem 2: Let $\{\mu_i\}$ be a countably infinite sequence of $p$-invariant ergodic measures on the circle whose normalized base-$p$ measures satisfy $h(\mu_i,\sigma_p)> 0$. Suppose that $\mu^\wedge$ is a joining of full entropy of $\{\mu_i\}$. Define $\Theta^n:\bbfT^\bbfN\to\bbfT$ by $\Theta^n(x)= x_1+\cdots+ x_n\pmod 1$. Then $h(\Theta^n\mu^\wedge, \sigma_p)$ tends to $\log p$ monotonically as $n$ tends to $\infty$.

Reviewed by Robert Cowen

Keywords: Furstenberg's conjecture; entropy for convolutions; ergodic measures; joining

Classification (MSC2000): 28D20 37A35

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