Weak Specification Property Research Articles

Lifting generic points

Abstract Let $(X,T)$ and $(Y,S)$ be two topological dynamical systems, where $(X,T)$ has the weak specification property. Let $\xi $ be an invariant measure on the product system $(X\times Y, T\times S)$ with marginals $\mu $ on X and $\nu $ on Y, with $\mu $ ergodic. Let $y\in Y$ be quasi-generic for $\nu $ . Then there exists a point $x\in X$ generic for $\mu $ such that the pair $(x,y)$ is quasi-generic for $\xi $ . This is a generalization of a similar theorem by T. Kamae, in which $(X,T)$ and $(Y,S)$ are full shifts on finite alphabets.

On n-scrambled sets

This paper investigates n-scrambled sets in the sense of distribution and Li-Yorke, focusing on the equivalence of n-scrambled sets and the existence of uncountable n-scrambled sets. For the equivalence, n-scrambled sets can imply m-scrambled sets in both senses for any 2≤m<n, but their inverses require certain conditions to hold. A method is provided for constructing an uncountable n-scrambled set in the distribution and Li-Yorke senses, which also verifies the equivalence between n-scrambled sets. For the existence, it is shown that distributionally n-scrambled tuples can be derived from the weak specification property (WSP), a 1-periodic and a non-1-periodic point. And the Li-Yorke (n+1)-scrambled tuple can be characterized by an n-scrambled tuple utilizing the SP. Finally, the uncountable Li-Yorke n-scrambled set is established based on the weak mixing, and the uncountable mean Li-Yorke n-scrambled set is induced by the shadowing and Devaney chaos.

The irregular set for maps with almost weak specification property has full metric mean dimension

Let (X,d) be a compact metric space, f:X→X be a continuous transformation with the almost weak specification property and φ:X→R be a continuous function. We consider the set (called the irregular set for φ) of points for which the Birkhoff average of φ does not exist and show that this set is either empty or carries full Bowen upper and lower metric mean dimension.

Garden of Eden and weakly periodic points for certain expansive actions of groups

AbstractWe present several applications of the weak specification property and certain topological Markov properties, recently introduced by Barbieri, García-Ramos, and Li [Markovian properties of continuous group actions: algebraic actions, entropy and the homoclinic group. Adv. Math.397 (2022), 52], and implied by the pseudo-orbit tracing property, for general expansive group actions on compact spaces. First we show that any expansive action of a countable amenable group on a compact metrizable space satisfying the weak specification and strong topological Markov properties satisfies the Moore property, that is, every surjective endomorphism of such dynamical system is pre-injective. This together with an earlier result of Li (where the strong topological Markov property is not needed) of the Myhill property [Garden of Eden and specification. Ergod. Th. &amp; Dynam. Sys.39 (2019), 3075–3088], which we also re-prove here, establishes the Garden of Eden theorem for all expansive actions of countable amenable groups on compact metrizable spaces satisfying the weak specification and strong topological Markov properties. We hint how to easily generalize this result even for uncountable amenable groups and general compact, not necessarily metrizable, spaces. Second, we generalize the recent result of Cohen [The large scale geometry of strongly aperiodic subshifts of finite type. Adv. Math.308 (2017), 599–626] that any subshift of finite type of a finitely generated group having at least two ends has weakly periodic points. We show that every expansive action of such a group having a certain Markov topological property, again implied by the pseudo-orbit tracing property, has a weakly periodic point. If it has additionally the weak specification property, the set of such points is dense.

Topological Weak Specification and Distributional Chaos on Noncompact Spaces

In this paper, we relate the topological definitions of specification property and distributional chaos defined for uniformly continuous self-maps on noncompact, nonmetrizable spaces. We prove that a uniformly continuous surjective self-map acting on a uniformly locally compact Hausdorff uniform space with topologically weak specification property and a pair of distal points is topologically distributionally chaotic of type 1. This extends the result due to Oprocha and Štefánková [2008]. As a consequence, we get that uniformly continuous surjective self-map on a uniformly locally compact totally bounded Hausdorff uniform space with topological shadowing, topological mixing, and a distal pair is topologically distributionally chaotic of type 1.

Specification and Shadowing Properties for Non-autonomous Systems

In this paper, various properties of non-autonomous dynamical systems having periodic shadowing property and local weak specification property are studied. The main theorem gives an interrelation among the shadowing property, periodic shadowing property and local weak specification property of an expansive non-autonomous system.

Expansive Actions with Specification on Uniform Spaces, Topological Entropy, and the Myhill Property

We prove that every expansive continuous action with the weak specification property of an amenable group G on a compact Hausdorff space X has the Myhill property, i.e., every pre-injective continuous self-mapping of X commuting with the action of G on X is surjective. This extends a result previously obtained by Hanfeng Li in the case when X is metrizable.

The weak specification property for geodesic flows on $\mathrm{CAT}(–1)$ spaces

We prove that the geodesic flow on a compact locally \mathrm{CAT}(–1) space has the weak specification property, and give various applications. We show that every Hölder potential on the space of geodesics has a unique equilibrium state. We establish the equidistribution of weighted periodic orbits and the large deviations principle for all such measures. The thermodynamic results are proved for the class of expansive flows with weak specification.

Distributional Chaos on Uniform Spaces

We introduce and study here the notion of distributional chaos on uniform spaces. We prove that if a uniformly continuous self-map of a uniform locally compact Hausdorff space has topological weak specification property then it admits a topologically distributionally scrambled set of type 3. This extends result due to Sklar and Smital (J Math Anal Appl 241:181–188, 2000). We also justify through examples necessity of the conditions in the hypothesis of the main result.

Weak specification and Baire Space

In this paper we compute the topological entropy and we consider the question of ω-chaos for shift spaces over countable alphabets. We show that if a shift space has the weak specification property, then it has ω-chaos, and we show that if, in addition, it has a countably infinite alphabet, then it has infinite topological entropy.

Topological and almost Borel universality for systems with the weak specification property

We show that systems with some specification properties are topologically or almost Borel universal, in the sense that any aperiodic subshift with lower entropy may be topologically or almost Borel embedded. This improves, with elementary tools, previous results of Quas and Soo [Ergodic universality of some topological dynamical systems. Trans. Amer. Math. Soc.368 (2016), 4137–4170].

Garden of Eden and specification

We establish a Garden of Eden theorem for expansive algebraic actions of amenable groups with the weak specification property, i.e. for any continuous equivariant map $T$ from the underlying space to itself, $T$ is pre-injective if and only if it is surjective. In particular, this applies to all expansive principal algebraic actions of amenable groups and expansive algebraic actions of $\mathbb{Z}^{d}$ with completely positive entropy.

Some weak specification properties and strongly mixing

In this paper, the authors first construct a dynamical system which is strongly mixing but has no weak specification property. Then the authors introduce two new concepts which are called the quasi-weak specification property and the semi-weak specification property in this paper, respectively, and the authors prove the equivalence of quasi-weak specification property, semi-weak specification property and strongly mixing.

Generic points of invariant measures for an amenable residually finite group actions with the weak specification property

We prove that every measure invariant for an amenable residually finite group action satisfying the weak specification property has a generic point. This extends the Sigmund theorem and completes the result of Ren.

Generic points for dynamical systems with average shadowing

We study the Besicovitch pseudometric D_B for compact dynamical systems. The set of generic points of ergodic measures turns out to be closed with respect to D_B. It is proved that the weak specification property implies the average asymptotic shadowing property and the latter property does not imply the former one nor the almost specification property. Furthermore an example of a proximal system with the average shadowing property is constructed. It is proved that to every invariant measure mu of a compact dynamical system one can associate a certain asymptotic pseudo orbit such that any point asymptotically tracing in average that pseudo orbit is generic for mu . A simple consequence of the theory presented is that every invariant measure has a generic point in a system with the asymptotic average shadowing property.

On entropy of dynamical systems with almost specification

We construct a family of shift spaces with almost specification and multiple measures of maximal entropy. This answers a question from Climenhaga and Thompson [Israel J. Math. 192 (2012), 785–817]. Elaborating on our examples we prove that sufficient conditions for every shift factor of a shift space to be intrinsically ergodic given by Climenhaga and Thompson are in some sense best possible; moreover, the weak specification property neither implies intrinsic ergodicity, nor follows from almost specification. We also construct a dynamical system with the weak specification property, which does not have the almost specification property. We prove that the minimal points are dense in the support of any invariant measure of a system with the almost specification property. Furthermore, if a system with almost specification has an invariant measure with non-trivial support, then it also has uniform positive entropy over the support of any invariant measure and cannot be minimal.

Shadowing, expansiveness and specification for C1-conservative systems

We prove that a C1-generic volume-preserving dynamical system (diffeomorphism or flow) has the shadowing property or is expansive or has the weak specification property if and only if it is Anosov. Finally, as in [10,27], we prove that the C1-robustness, within the volume-preserving context, of the expansiveness property and the weak specification property, imply that the dynamical system (diffeomorphism or flow) is Anosov.

Minimal sets and $\omega$-chaos in expansive systems with weak specification property

In this paper, we first discuss minimal sets in the compact dynamical system with weak specification property. On the basis of this discussion, we show that an expansive system with weak specification property displays a stronger form of $\omega$-chaos.

Dynamical systems generating large sets of probability distribution functions

For a topological dynamical system (X,f) we consider the structure of the set F(f) of asymptotic distributions of the distances between pairs of trajectories. If f has the weak specification property then F(f) is closed and convex, and it can contain all nondecreasing functions [0,diamX]→[0,1]. We discuss relations to distributional chaos.

Irregular sets for ratios of Birkhoff averages are residual

It follows from Birkhoff's Ergodic Theorem that the irregular set of points for which the Birkhff averages of a given continuous function diverge has zero measure with respect to any finite invariant measure. In strong contrast, for systems with the weak specification property, we show here that if the irregular set is nonempty, then it is residual. This includes topologically transitive topological Markov chains, sofic shifts and more generally shifts with the specfication property. We consider also the more general case of ratios of Birkhoff averages of continuous functions and the case when the set of accumulation points of the ratios of Birkhoff averages is a prescribed closed interval. Finally, we give an application of our work to the pointwise dimension of a Gibbs measure on a repeller of a conformal map.