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Entropy of convolutions on the circle. (English) Zbl 0940.28015

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 \(\Theta^n:\mathbb{T}^\mathbb{N}\to\mathbb{T}\) 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 \(\infty\).

MSC:

28D20 Entropy and other invariants
37A35 Entropy and other invariants, isomorphism, classification in ergodic theory
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