## 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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