Dynamics of metrics in measure spaces and their asymptotic invariants (original) (raw)
Related papers
Geometry and dynamics of admissible metrics in measure spaces
Central European Journal of Mathematics, 2013
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 ɛ-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 ɛ-entropy of the averages of some (and hence any) admissible metric over its trajectory is uniformly bounded.
Bulletin of the Brazilian Mathematical Society, New Series, 2021
We investigate in this work some situations where it is possible to estimate or determine the upper and the lower q-generalized fractal dimensions D ± µ (q), q ∈ R, of invariant measures associated with continuous transformations over compact metric spaces. In particular, we present an alternative proof of Young's Theorem [31] for the generalized fractal dimensions of the Bowen-Margulis measure associated with a C 1+α-Axiom A system over a two-dimensional compact Riemannian manifold M. We also present estimates for the generalized fractal dimensions of an ergodic measure for which Brin-Katok's Theorem is satisfied punctually, in terms of its metric entropy. Furthermore, for expansive homeomorphisms (like C 1-Axiom A systems), we show that the set of invariant measures such that D + µ (q) = 0 (q ≥ 1), under a hyperbolic metric, is generic (taking into account the weak topology). We also show that for each s ∈ [0, 1), D + µ (s) is bounded above, up to a constant, by the topological entropy, also under a hyperbolic metric. Finally, we show that, for some dynamical systems, the metric entropy of an invariant measure is typically zero, settling a conjecture posed by Sigmund in [25] for Lipschitz transformations which satisfy the specification property. Key words and phrases. Expansive homeomorphisms, Hausdorff dimension, packing dimension, invariant measures, generalized fractal dimensions, dynamical systems with specification * Work partially supported by CIENCIACTIVA C.G. 176-2015 † Work partially supported by FAPEMIG (a Brazilian government agency; Universal Project 001/17/CEX-APQ-00352-17) popular of all, the Hausdorff dimension, introduced in 1919 by Hausdorff, which gives a notion of size useful for distinguishing between sets of zero Lebesgue measure. Unfortunately, the Hausdorff dimension of relatively simple sets can be very hard to calculate; besides, the notion of Hausdorff dimension is not completely adapted to the dynamics per se (for instance, if Z is a periodic orbit, then its Hausdorff dimension is zero, regardless to whether the orbit is stable, unstable, or neutral). This fact led to the introduction of other characteristics for which it is possible to estimate the size of irregular sets. For this reason, some of these quantities were also branded as "dimensions" (although some of them lack some basic properties satisfied by Hausdorff dimension, such as σ-stability; see [12]). Several good candidates were proposed, such as the correlation, information, box counting and entropy dimensions, among others. Thus, in order to obtain relevant information about the dynamics, one should consider not only the geometry of the measurable set Z ⊂ X (where X is some Borel measurable space), but also the distribution of points on Z under f (which is assumed to be a measurable transformation). That is, one should be interested in how often a given point x ∈ Z visits a fixed subset Y ⊂ Z under f. If µ is an ergodic measure for which µ(Y) > 0, then for a typical point x ∈ Z, the average number of visits is equal to µ(Y). Thus, the orbit distribution is completely determined by the measure µ. On the other hand, the measure µ is completely specified by the distribution of a typical orbit. This fact is widely used in the numerical study of dynamical systems where the distributions are, in general, non-uniform and have a clearly visible fine-scaled interwoven structure of hot and cold spots, that is, regions where the frequency of visitations is either much greater than average or much less than average respectively.
Mathematische Nachrichten
In this work, we show that if f is a uniformly continuous map defined over a Polish metric space, then the set of f‐invariant measures with zero metric entropy is a set (in the weak topology). In particular, this set is generic if the set of f‐periodic measures is dense in the set of f‐invariant measures. This settles a conjecture posed by Sigmund (Trans. Amer. Math. Soc. 190 (1974), 285–299), which states that the metric entropy of an invariant measure of a topological dynamical system that satisfies the periodic specification property is typically zero. We also show that if X is compact and if f is an expansive or a Lipschitz map with a dense set of periodic measures, typically the lower correlation entropy for is equal to zero. Moreover, we show that if X is a compact metric space and if f is an expanding map with a dense set of periodic measures, then the set of invariant measures with packing dimension, upper rate of recurrence and upper quantitative waiting time indicator equa...
On Asymptotic Behavior of Metric Dynamical Systems
2015
In this paper we shall study the asymptotic behavior of metric dynamical systems when the time domain is any locally compact topological group. We investigate some new properties of ergodic, mixing, and weakly mixing metric dynamical system.
Some results on the entropy of non-autonomous dynamical systems
Dynamical Systems-an International Journal, 2015
In this paper we advance the entropy theory of discrete nonautonomous dynamical systems that was initiated by Kolyada and Snoha in 1996. The first part of the paper is devoted to the measure-theoretic entropy theory of general topological systems. We derive several conditions guaranteeing that an initial probability measure, when pushed forward by the system, produces an invariant measure sequence whose entropy captures the dynamics on arbitrarily fine scales. In the second part of the paper, we apply the general theory to the nonstationary subshifts of finite type, introduced by Fisher and Arnoux. In particular, we give sufficient conditions for the variational principle, relating the topological and measure-theoretic entropy, to hold.
Packing topological entropy for amenable group actions
Ergodic Theory and Dynamical Systems, 2021
Packing topological entropy is a dynamical analogy of the packing dimension, which can be viewed as a counterpart of Bowen topological entropy. In the present paper we give a systematic study of the packing topological entropy for a continuous G-action dynamical system (X,G)(X,G)(X,G) , where X is a compact metric space and G is a countable infinite discrete amenable group. We first prove a variational principle for amenable packing topological entropy: for any Borel subset Z of X, the packing topological entropy of Z equals the supremum of upper local entropy over all Borel probability measures for which the subset Z has full measure. Then we obtain an entropy inequality concerning amenable packing entropy. Finally, we show that the packing topological entropy of the set of generic points for any invariant Borel probability measure mu\mu mu coincides with the metric entropy if either mu\mu mu is ergodic or the system satisfies a kind of specification property.
Toward the History of Dynamical Entropy: Comparing Two Definitions
Journal of Mathematical Sciences, 2016
We prove that for ergodic automorphisms of a Lebesgue space, the definition of the measuretheoretic entropy suggested in the master thesis by D. Arov (1957) and remained unpublished and the well-known definition of Sinai (1959) reduce to each other, while in general this is not the case. Bibliography: 11 titles.
arXiv (Cornell University), 2019
We show in this work that the upper and the lower generalized fractal dimensions Dpmmu(q)D^{\pm}_{\mu}(q)Dpmmu(q), for each qinmathbbRq\in\mathbb{R}qinmathbbR, of an ergodic measure associated with an invertible bi-Lipschitz transformation over a Polish metric space are equal, respectively, to its packing and Hausdorff dimensions. This is particularly true for hyperbolic ergodic measures associated with C1+alphaC^{1+\alpha}C1+alpha-diffeomorphisms of smooth compact Riemannian manifolds, from which follows an extension of Young's Theorem (Young, L., S. Dimension, entropy and Lyapunov exponents. Ergodic Theory and Dynamical Systems, 2(1):109-124, 1982). Analogous results are obtained for expanding systems. Furthermore, for expansive homeomorphisms (like C1C^1C1-Axiom A systems), we show that the set of invariant measures with zero correlation dimension, under a hyperbolic metric, is generic (taking into account the weak topology). We also show that for each sge0s\ge 0sge0, D+mu(s)D^{+}_{\mu}(s)D+mu(s) is bounded above, up to a constant, by the topological entropy, also under a hyperbolic metric.