scholarly journals Lyapunov exponents for expansive homeomorphisms

2020 ◽  
Vol 63 (2) ◽  
pp. 413-425
Author(s):  
M. J. Pacifico ◽  
J. L. Vieitez
Keyword(s):  
Dynamical Systems ◽  
Lyapunov Exponents ◽  
Metric Space ◽  
Metric Spaces ◽  
Invariant Sets ◽  
Dimension Theory ◽  
Compact Set ◽  
Compact Metric Space ◽  
Characteristic Exponents ◽  
Expansive Homeomorphism

AbstractWe address the problem of defining Lyapunov exponents for an expansive homeomorphism f on a compact metric space (X, dist) using similar techniques as those developed in Barreira and Silva [Lyapunov exponents for continuous transformations and dimension theory, Discrete Contin. Dynam. Sys.13 (2005), 469–490]; Kifer [Characteristic exponents of dynamical systems in metric spaces, Ergod. Th. Dynam. Sys.3 (1983), 119–127]. Under certain conditions on the topology of the space X where f acts we obtain that there is a metric D defining the topology of X such that the Lyapunov exponents of f are different from zero with respect to D for every point x ∈ X. We give an example showing that this may not be true with respect to the original metric dist. But expansiveness of f ensures that Lyapunov exponents do not vanish on a Gδ subset of X with respect to any metric defining the topology of X. We define Lyapunov exponents on compact invariant sets of Peano spaces and prove that if the maximal exponent on the compact set is negative then the compact is an attractor.

scholarly journals Characteristic exponents of dynamical systems in metric spaces

1983 ◽  
Vol 3 (1) ◽  
pp. 119-127 ◽  
Author(s):  
Yuri Kifer
Keyword(s):  
Dynamical Systems ◽  
Metric Spaces ◽  
Invariant Sets ◽  
Characteristic Exponents ◽  
Smooth Case

AbstractWe introduce for dynamical systems in metric spaces some numbers which in the case of smooth dynamical systems turn out to be the maximal and the minimal characteristic exponents. These numbers have some properties similar to the smooth case. Analogous quantities are defined also for invariant sets.

scholarly journals Sensitivity for topologically double ergodic dynamical systems

2021 ◽  
Vol 6 (10) ◽  
pp. 10495-10505
Author(s):  
Risong Li ◽  
◽  
Xiaofang Yang ◽  
Yongxi Jiang ◽  
Tianxiu Lu ◽  
...  
Keyword(s):  
Dynamical Systems ◽  
Metric Space ◽  
Inverse Limit ◽  
Compact Metric Space ◽  
Limit Space ◽  
Factor Map ◽  
Continuous Surjection ◽  
Inverse Limit Space

<abstract><p>As a stronger form of multi-sensitivity, the notion of ergodic multi-sensitivity (resp. strongly ergodically multi-sensitivity) is introduced. In particularly, it is proved that every topologically double ergodic continuous selfmap (resp. topologically double strongly ergodic selfmap) on a compact metric space is ergodically multi-sensitive (resp. strongly ergodically multi-sensitive). And for any given integer $ m\geq 2 $, $ f $ is ergodically multi-sensitive (resp. strongly ergodically multi-sensitive) if and only if so is $ f^{m} $. Also, it is shown that if $ f $ is a continuous surjection, then $ f $ is ergodically multi-sensitive (resp. strongly ergodically multi-sensitive) if and only if so is $ \sigma_{f} $, where $ \sigma_{f} $ is the shift selfmap on the inverse limit space $ \lim\limits_{\leftarrow}(X, f) $. Moreover, it is proved that if $ f:X\rightarrow X $ (resp. $ g:Y\rightarrow Y $) is a map on a nontrivial metric space $ (X, d) $ (resp. $ (Y, d') $), and $ \pi $ is a semiopen factor map between $ (X, f) $ and $ (Y, g) $, then the ergodic multi-sensitivity (resp. the strongly ergodic multi-sensitivity) of $ g $ implies the same property of $ f $.</p></abstract>

Exponential global attractors for semigroups in metric spaces with applications to differential equations

2011 ◽  
Vol 31 (6) ◽  
pp. 1641-1667 ◽  
Author(s):  
ALEXANDRE N. CARVALHO ◽  
JAN W. CHOLEWA
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Global Attractor ◽  
Metric Space ◽  
Metric Spaces ◽  
Global Attractors ◽  
Invariant Sets ◽  
Bounded Subset ◽  
Partial Differential ◽  
Unstable Manifolds

AbstractIn this article semigroups in a general metric space V, which have pointwise exponentially attracting local unstable manifolds of compact invariant sets, are considered. We show that under a suitable set of assumptions these semigroups possess strong exponential dissipative properties. In particular, there exists a compact global attractor which exponentially attracts each bounded subset of V. Applications of abstract results to ordinary and partial differential equations are given.

Finite powers of strong measure zero sets

1999 ◽  
Vol 64 (3) ◽  
pp. 1295-1306 ◽  
Author(s):  
Marion Scheepers
Keyword(s):  
Metric Space ◽  
Metric Spaces ◽  
Measure Zero ◽  
General Context ◽  
Partition Relation ◽  
Compact Metric Space ◽  
Totally Bounded ◽  
Zero Sets ◽  
Strong Measure ◽  
Strong Measure Zero

AbstractIn a previous paper—[17]—we characterized strong measure zero sets of reals in terms of a Ramseyan partition relation on certain subspaces of the Alexandroff duplicate of the unit interval. This framework gave only indirect access to the relevant sets of real numbers. We now work more directly with the sets in question, and since it costs little in additional technicalities, we consider the more general context of metric spaces and prove:1. If a metric space has a covering property of Hurewicz and has strong measure zero, then its product with any strong measure zero metric space is a strong measure zero metric space (Theorem 1 and Lemma 3).2. A subspace X of a σ-compact metric space Y has strong measure zero if, and only if, a certain Ramseyan partition relation holds for Y (Theorem 9).3. A subspace X of a σ-compact metric space Y has strong measure zero in all finite powers if, and only if, a certain Ramseyan partition relation holds for Y (Theorem 12).Then 2 and 3 yield characterizations of strong measure zeroness for σ-totally bounded metric spaces in terms of Ramseyan theorems.

scholarly journals Bowen’s equation in the non-uniform setting

2010 ◽  
Vol 31 (4) ◽  
pp. 1163-1182 ◽  
Author(s):  
VAUGHN CLIMENHAGA
Keyword(s):  
Hausdorff Dimension ◽  
Lyapunov Exponents ◽  
Metric Space ◽  
Periodic Points ◽  
Topological Pressure ◽  
Conformal Map ◽  
Arbitrary Subset ◽  
Compact Metric Space ◽  
Symbolic Space ◽  
Contraction Condition

AbstractWe show that Bowen’s equation, which characterizes the Hausdorff dimension of certain sets in terms of the topological pressure of an expanding conformal map, applies in greater generality than has been heretofore established. In particular, we consider an arbitrary subset Z of a compact metric space and require only that the lower Lyapunov exponents be positive on Z, together with a tempered contraction condition. Among other things, this allows us to compute the dimension spectrum for Lyapunov exponents for maps with parabolic periodic points, and to relate the Hausdorff dimension to the topological entropy for arbitrary subsets of symbolic space with the appropriate metric.

scholarly journals Countable contraction mappings in metric spaces: invariant sets and measure

2014 ◽  
Vol 12 (4) ◽  
Author(s):  
María Barrozo ◽  
Ursula Molter
Keyword(s):  
Invariant Measure ◽  
Metric Space ◽  
Metric Spaces ◽  
Complete Metric Space ◽  
Invariant Set ◽  
Invariant Sets ◽  
Countable Number ◽  
Contraction Mappings ◽  
Classic Result ◽  
System F

AbstractWe consider a complete metric space (X, d) and a countable number of contraction mappings on X, F = {F i: i ∈ ℕ}. We show the existence of a smallest invariant set (with respect to inclusion) for F. If the maps F i are of the form F i(x) = r i x + b i on X = ℝd, we prove a converse of the classic result on contraction mappings, more precisely, there exists a unique bounded invariant set if and only if r = supi r i is strictly smaller than 1.Further, if ρ = {ρ k}k∈ℕ is a probability sequence, we show that if there exists an invariant measure for the system (F, ρ), then its support must be precisely this smallest invariant set. If in addition there exists any bounded invariant set, this invariant measure is unique, even though there may be more than one invariant set.

scholarly journals Notions of relative ubiquity for invariant sets of relational structures

1990 ◽  
Vol 55 (3) ◽  
pp. 948-986 ◽  
Author(s):  
Paul Bankston ◽  
Wim Ruitenburg
Keyword(s):  
Probability Measure ◽  
Metric Space ◽  
Measure Space ◽  
Invariant Sets ◽  
Natural Numbers ◽  
Compact Metric Space ◽  
Linear Orderings ◽  
Relational Structures ◽  
Probability Measure Space

AbstractGiven a finite lexicon L of relational symbols and equality, one may view the collection of all L-structures on the set of natural numbers ω as a space in several different ways. We consider it as: (i) the space of outcomes of certain infinite two-person games; (ii) a compact metric space; and (iii) a probability measure space. For each of these viewpoints, we can give a notion of relative ubiquity, or largeness, for invariant sets of structures on ω. For example, in every sense of relative ubiquity considered here, the set of dense linear orderings on ω is ubiquitous in the set of linear orderings on ω.

Directional entropy of ℤ+k-actions

2015 ◽  
Vol 16 (01) ◽  
pp. 1650004 ◽  
Author(s):  
Jinlian Zhang ◽  
Wenda Zhang
Keyword(s):  
Dynamical System ◽  
Variational Principle ◽  
Lyapunov Exponents ◽  
Riemannian Manifolds ◽  
Metric Space ◽  
Compact Metric Space ◽  
Finite Graphs ◽  
Nonautonomous Dynamical System

In this paper, topological and measure-theoretic directional entropies are investigated for [Formula: see text]-actions. Let [Formula: see text] be a [Formula: see text]-action on a compact metric space. For each ray [Formula: see text] in [Formula: see text] we introduce a notion of positive expansivity for [Formula: see text] along [Formula: see text]. We apply the technique of “coding” which was given by Boyle and Lind in [1] to show that these directional entropies are both continuous at positively expansive directions. We relate the directional entropies of a [Formula: see text]-action at a ray [Formula: see text] to the entropies of a nonautonomous dynamical system which induced by the compositions of a sequence of maps along [Formula: see text]. And hence the variational principle relating topological and measure-theoretic directional entropies is given at positively expansive directions. Applying some known results relating entropies and other invariants (such as preimage entropies, degrees and Lyapunov exponents), we obtain the formulas of directional entropies for some classic examples, such as the [Formula: see text]-subshift actions on [Formula: see text], [Formula: see text]-actions on finite graphs and certain smooth [Formula: see text]-actions on Riemannian manifolds.

scholarly journals ON BUCK’S UNIFORM DISTRIBUTION IN COMPACT METRIC SPACES

2013 ◽  
Vol 56 (1) ◽  
pp. 61-66
Author(s):  
Milan Paštéka
Keyword(s):  
Uniform Distribution ◽  
Metric Space ◽  
Existence Result ◽  
Metric Spaces ◽  
Theoretic Approach ◽  
Compact Metric Space ◽  
Compact Metric Spaces

ABSTRACT We define uniform distribution in compact metric space with respect to the Buck’s measure density originated in [Buck, R. C.: The measure theoretic approach to density, Amer. J. Math. 68 (1946), 560-580]. Weyl’s criterion is derived. This leads to an existence result.

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