Entropy of convolutions on the circle

• 成果类型: Article
• 署名作者: Lindenstrauss, E; Meiri, D; Peres, Y
• 刊物名称: ANNALS OF MATHEMATICS
• ISSN/ISSBN: 0003-486X
• DOI: 10.2307/121075
• 发表日期: 1999
• 页码: 871-904
• 关键词: normal numbers; invariant-measures

摘要

Given ergodic p-invariant measures {mu(i)} on the 1-torus T = R/Z, we give a sharp condition on their entropies, guaranteeing that the entropy of the convolution mu(1) *...* mu(n) converges to log p. We also prove a variant of this result for joinings of full entropy on T-N. In conjunction with a method of Host, this yields the following. Denote sigma(q)(x) = qx (mod 1). Then for every p-invariant ergodic mu with positive entropy, 1/N Sigma(n=0)(N-1) sigma(cn)mu converges weak* to Lebesgue measure as N --> infinity, 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(2k).) 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 (Si) of p-invariant closed subsets of T, if Sigma dim(H)(S-i)/\ log dim(H)(S-i)\ = infinity, then dim(H)(S-1 +...+ S-n) --> 1.