Geometry and dynamics of admissible metrics in measure spaces

Abstract

We study a wide class of metrics in a Lebesgue space, namely the class of so-called admissible metrics. We consider the cone of admissible metrics, introduce a special norm in it, prove compactness criteria, define the ε-entropy of a measure space with an admissible metric, etc. These notions and related results are applied to the theory of transformations with invariant measure; namely, we study the asymptotic properties of orbits in the cone of admissible metrics with respect to a given transformation or a group of transformations. The main result of this paper is a new discreteness criterion for the spectrum of an ergodic transformation: we prove that the spectrum is discrete if and only if the ε-entropy of the averages of some (and hence any) admissible metric over its trajectory is uniformly bounded. MSC:37A05, 11J83, 37C85

Bounded complexity, mean equicontinuity and discrete spectrum

“…We would like to thank Bryna Kra for the helpful discussion on the subject, specially on Theorem 4.7. We also would like to thank Nhan-Phu Chung for bringing [28] into our attention. W. Huang was partially supported by NNSF of China (11431012,11731003), J. Li was partially supported by NNSF of China (11771264) and…”

On The Possible Growth Rate of a Scaling Entropy Sequence

“…Therefore, this cone contains all semimetrics dominated by finite sums of cut semimetrics. It remains to recall that every bounded semimetric can be approximated in the m-norm by such semimetrics, and every admissible summable semimetric can be approximated in the m-norm by bounded ones (for instance, by its cut-offs, see [18,Lemma 2.16]). Thus, for every summable admissible metric ρ, the cone M ρ coincides with the set of all summable admissible semimetrics on (X, μ).…”

“…Let h = {h n } be a nondecreasing subadditive sequence. It is known (see, e.g., [18]) that a system has bounded scaling entropy sequences if and only if it has a discrete spectrum (it is worth noting that such systems also can be realized on BratteliVershik diagrams, though we do not use this fact in this paper). Hence we may assume that the sequence h is unbounded.…”

Виртуальная Непрерывность Измеримых Функций Многих Переменных И Ее Приложения

“…Свойства допустимых полуметрик и метрик подробно изучены в рабо-тах [10], [24]. В частности, там приведен ряд равносильных определений этого понятия.…”