Entropy of convolutions on the circle
Entropy of convolutions on the circle
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圆上卷积的熵
DOI：
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发表时间：
1999
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通讯作者：
Y. Peres
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文献类型：
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作者：
E. Lindenstrauss;D. Meiri;Y. Peres
给定1-环面上的遍历p-不变测度f ig，我们给出了它们的熵的一个严格条件，保证卷积的熵收敛于logp.我们还证明了这个结果的一个变种的连接的全熵。结合Host的方法，这将产生以下结果。表示aeq（x）= qx（mod 1）。则对每一个具有正熵的p-不变遍历n，1 N P Ni 1 n = 0 aecn n收敛于弱l to Lebesgue测度为N i! 1，在fckg. (For例如，当p = 10，ck = 2 k +6 k或ck = 2 2 k时，条件成立。这扩展了约翰逊和鲁道夫的结果，他们认为当p和q是乘法独立的序列ck = q k。我们还得到了关于和集的Hausdorfi维数的推论：对任意p-不变闭子集序列fSig，i f PdimH（Si）= jlogdimH（Si）j = 1，则dimH（S1 + Sn）i! 1.
Given ergodic p-invariant measures f„ig on the 1-torus = = ,w e give a sharp condition on their entropies, guaranteeing that the entropy of the convolution „1⁄¢¢¢⁄„n converges to logp. We also prove a variant of this result for joinings of full entropy on . In conjunction with a method of Host, this yields the following. Denote aeq(x )= qx (mod 1). Then for every p-invariant ergodic „ with positive entropy, 1 N P Ni1 n=0 aecn „ converges weak ⁄ to Lebesgue measure as N i! 1 , under a certain mild combinatorial condition onfckg. (For instance, the condition is satisfled if p = 10 and ck =2 k +6 k or ck =2 2 k .) This extends a result of Johnson and Rudolph, who considered the sequence ck = q k when p and q are multiplicatively independent. We also obtain the following corollary concerning Hausdorfi dimension of sum sets: For any sequence fSig of p-invariant closed subsets of ,i f P dimH(Si)=j log dimH(Si)j =1, then dimH(S1 +¢¢¢+Sn)i! 1.