These pages are not updated anymore. For the current production of this journal, please refer to http://www.jstor.org/journals/0003486x.html.
Annals of Mathematics, II. Series Vol. 149, No. 3, pp. 871904 (1999) 

Entropy of convolutions on the circleElon Lindenstrauss, David Meiri and Yuval PeresReview 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 Full text of the article:
Electronic fulltext finalized on: 8 Sep 2001. This page was last modified: 21 Jan 2002.
© 2001 Johns Hopkins University Press
