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Scaling entropy and automorphisms with pure point spectrum. (English. Russian original) Zbl 1251.37009

St. Petersbg. Math. J. 23, No. 1, 75-91 (2012); translation from Algebra Anal. 23, No. 1, 111-135 (2011).
The author introduces the notion of a scaling sequence for the \(\varepsilon\)-entropies of a measure-preserving transformation on a Lebesgue space and establishes the following theorem:
A measure-preserving transformation has a discrete spectrum if and only if the scaling sequence for the \(\varepsilon\)-entropies with respect to the averages of an admissible metric is bounded.
A metric \(\rho\) is said to be admissible if \[ \displaystyle \int_X\int_X \rho(x,y)d\mu(x)d\mu(y)<\infty. \] The notion of a scaling sequence is defined as follows: Let \(T\) be a measure-preserving transformation on a Lebesgue space \((X,\mathcal{B},\mu)\) and \(\rho\) a metric on \(X\). The class of scaling sequences for \(T\), denoted by \(H_{\rho,\varepsilon}(T)\), consists of monotone increasing sequences of positive numbers \(\{c_n\}_{n\in \mathbb{N}}\) such that \[ 0<\liminf\left(\frac{H_{\rho_n,\varepsilon}(\mu)}{c_n}\right)< \limsup\left (\frac{H_{\rho_n,\varepsilon}(\mu)}{c_n}\right)<\infty, \] where the metric \(\rho_n\) is defined by \[ \rho_n=\frac{1}{n}\sum_{k=0}^{n-1}\rho \circ {(T\times T)}^k \] and \[ H_{\rho_n,\varepsilon}(\mu)=\inf\left\{\ln(k): \text{there exist }X' \text{ with } \mu(X') > 1-\varepsilon \text{ and } (x_i)_{i=1}^{k} \subset X' \text{ with } X' \subset \bigcup_{i=1}^{k}B_{\rho_n}(x_i,\varepsilon)\right\}, \] \(B_{\rho_n}(x_i,\varepsilon)\) being an \(\varepsilon\)-ball centered at \(x_i\) with respect to the metric \(\rho_n\).

MSC:

37A35 Entropy and other invariants, isomorphism, classification in ergodic theory
28D05 Measure-preserving transformations
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References:

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