Expansive Dynamical Systems Research Articles

Synchronizing dynamical systems: Their groupoids and 𝐶*-algebras

Building on work of Ruelle and Putnam in the Smale space case, Thomsen defined the homoclinic and heteroclinic C ∗ C^\ast -algebras for an expansive dynamical system. In this paper we define a class of expansive dynamical systems, called synchronizing dynamical systems, that exhibit hyperbolic behavior almost everywhere. Synchronizing dynamical systems generalize Smale spaces (and even finitely presented systems). Yet they still have desirable dynamical properties such as having a dense set of periodic points. We study various C ∗ C^\ast -algebras associated with a synchronizing dynamical system. Among other results, we show that the homoclinic algebra of a synchronizing system contains an ideal which behaves like the homoclinic algebra of a Smale space.

Expansive actions with specification of sofic groups, strong topological Markov property, and surjunctivity

A dynamical system is a pair (X,G), where X is a compact metrizable space and G is a countable group acting by homeomorphisms of X. An endomorphism of (X,G) is a continuous selfmap of X which commutes with the action of G. One says that a dynamical system (X,G) is surjunctive provided that every injective endomorphism of (X,G) is surjective (and therefore is a homeomorphism). We show that when G is sofic, every expansive dynamical system (X,G) with nonnegative sofic topological entropy and satisfying the weak specification and the strong topological Markov properties, is surjunctive.

Kinematic N-expansive continuous dynamical systems

Expansiveness has been used to study dynamic systems and has been developed for various forms of expansiveness. In this paper, we introduce the concept of kinematic N-expansiveness for flows on a C∞ compact connected manifold M, which is an extension of N-expansive homeomorphisms. We prove that if a vector field X on M is C1 robustly kinematic N-expansive, then it is quasi-Anosov. Furthermore, we consider the divergence-free vector fields and Hamiltonian systems with the kinematic N-expansive property; then, we study their robustness.

Expansive systems on lattices

We study expansive dynamical systems in the setting of distributive lattices and their automorphisms, the usual notion of expansiveness for a homeomorphism of a compact metric space being the particular case when the lattice is the topology of the phase space ordered by inclusion and the automorphism the one induced by the homeomorphism, mapping open sets to open sets. We prove in this context generalizations of Mañé's Theorem and Utz's Theorem about the finite dimensionality of the phase space of an expansive system, and the finiteness of a space supporting a positively expansive homeomorphism, respectively. We also discuss the notion of entropy in this setting calculating it for the non-Hausdorff shifts.

On periodic shadowing, transitivity, chain mixing and expansivity in uniform dynamical systems

In this paper we extend some results on the notions such as Expansive, Pseudo Orbit Tracing Property (P.O.T.P.), Chain Transitive, Periodic Shadowing, Chain Recurrent. We prove that if an expansive dynamical system (X,f) on compact uniform space has P.O.T.P., then it has periodic shadowing. If a continuous self map f on a compact uniform space has finite shadowing, then f has P.O.T.P. We find that a dynamical system (X,f) on compact uniform space has shadowing property if (X,f) has periodic shadowing provided f is expansive. If f is chain mixing on a compact uniform space (X,U), then fn is chain transitive for each n ≥ 1. If f has the periodic shadowing then fn has periodic shadowing for all n > 1. The periodic shadowing is invariant of topological conjugacy provided that the conjugacy and its inverse are Lipschitz.

Extensions of expansive dynamical systems

<p style='text-indent:20px;'>We characterize and describe the extensions of expansive and Ano- sov homeomorphisms on compact spaces. As an application we obtain a stability result for extensions of Anosov systems, and show a construction that embeds any expansive system inside an expansive system having the shadowing property for the pseudo orbits of the original space.

Expansive properties of induced dynamical systems

For a given metric space X, we consider the set of all normal fuzzy sets on X, denoted by F(X). In this work, we study expanding, positively expansive and weakly positively expansive dynamical systems (X,f) and how they are reflected in the dynamical system (F(X),fˆ), where fˆ is the Zadeh's extension of f and F(X) has one of the following metrics: the levelwise metric, the endograph metric, the sendograph metric and the Skorokhod metric. We mainly show that if we consider the following conditions:(i)(X,f) is positively expansive (resp. expanding);(ii)(K(X),f‾) is positively expansive (resp. expanding);(iii)(F∞(X),fˆ) is positively expansive (resp. expanding);(iv)(F0(X),fˆ) is positively expansive (resp. expanding). Then (iv)⇒ (iii) ⇔ (ii) ⇒ (i). For expanding dynamical systems, we present a compact metric space and a locally compact metric space to show that (i) ⇏ (ii) and (iii) ⇏ (iv), respectively. For positively expansive dynamical systems, there is a compact metric space satisfying that (i) ⇏ (ii), but we don't know if (iii) ⇒ (iv).

A Study of Equicontinuous Maps On Uniform Spaces G–Spaces By

In this paper we shall study some new properties of equicontinuous maps on uniform G–Spaces. Here the phase space consider as a uniform space. Also we show the relationship among the equicontinuous maps with the distal dynamical system and expansive dynamical system.

Continuum-wise expansive homoclinic classes for robust dynamical systems

In the study, we consider continuum-wise expansiveness for the homoclinic class of a kind of C^{1}-robustly expansive dynamical system. First, we show that if the homoclinic class H(p, f), which contains a hyperbolic periodic point p, is R-robustly continuum-wise expansive, then it is hyperbolic. For a vector field, if the homoclinic class H(gamma , X) does not include singularities and is R-robustly continuum-wise expansive, then it is hyperbolic.

Specification and Thermodynamic Properties of Topological Time-Dependent Dynamical Systems

This paper discusses the thermodynamic properties for certain time-dependent dynamical systems. In particular, we are interested in time-dependent dynamical systems with the specification property. We show that each time-dependent dynamical system given by a sequence of surjective continuous self maps of a compact metric space with the specification property has positive topological entropy and all points are entropy point. In particular, it is proved that these systems are topologically chaotic. We will treat the dynamics of uniformly Ruelle-expanding time-dependent dynamical systems on compact metric spaces and provide some sufficient conditions that these systems have the specification property. Consequently, we conclude that these systems have positive topological entropy. This extends a result of Kawan \cite{K2}, corresponding to the case when the expanding maps are smooth, to the more general case of expanding maps. Additionally, we study the topological pressure of time-dependent dynamical systems. We obtain conditions under which the topological entropy and topological pressure of any continuous potential can be computed as a limit at a definite size scale. Finally, we study the Lipschitz regularity of the topological pressure function for expansive (uniformly Ruelle-expanding) time-dependent dynamical systems on compact metric spaces.

Topologically sensitive dynamical systems

In this paper we introduce a notion of sensitivity for topological dynamical systems and show some of its basic features and relation to dynamical properties such as transitivity and weak mixing. Finally, we will restrict our attention to the related class of weakly positively expansive dynamical systems.

Periodic shadowing and standard shadowing property

In this paper we study the periodic shadowing property and show that expansive dynamical systems that have the pseudo-orbit tracing property (POTP) also have the periodic shadowing. In addition, the convers is proved for chain transitive systems.

On specification and measure expansiveness

We relate the local specification and periodic shadowing properties. We also clarify the relation between local weak specification and local specification if the system is measure expansive. The notion of strong measure expansiveness is introduced, and an example of a non-expansive systems with the strong measure expansive property is given. Moreover, we find a family of examples with the $N$-expansive property, which are not strong measure expansive. We finally show a spectral decomposition theorem for strong measure expansive dynamical systems with shadowing.

Expansive homeomorphisms on non-Hausdorff spaces

We study expansive dynamical systems from the viewpoint of general topology. We introduce the notions of orbit and refinement expansivity on topological spaces extending expansivity in the compact metric setting. Examples are given on non-Hausdorff compact spaces. Topological and dynamical properties are studied in relation to separability axioms, metrizability, uniform expansivity, asymptotic points and positive expansivity. We also prove that every refinement expansive homeomorphism on a T1 space, in particular every expansive homeomorphism in the usual sense, is conjugate to a subshift with a possibly non-Hausdorff topology on the symbols.

On intrinsic ergodicity and weakenings of the specification property

Since seminal work of Bowen [2], it has been known that the specification property implies various useful properties about an expansive topological dynamical system, among them uniqueness of the measure of maximal entropy (often referred to as intrinsic ergodicity). Weakenings of the specification property have been defined and profitably applied in various works such as [6,9,11,16,17].It has been an open question (see p. 798 of [4]) whether two of these properties, which we here call almost specification and non-uniform specification, imply intrinsic ergodicity for expansive topological systems. We answer this question negatively by exhibiting examples of subshifts with multiple measures of maximal entropy with disjoint support which have non-uniform specification with any gap function f(n)=O(ln⁡n) or almost specification with any mistake function g(n)≥4. We also show some results in the opposite direction, showing that subshifts with non-uniform specification with gap function f(n)=o(ln⁡n) or almost specification with mistake function g(n)=1 cannot have multiple measures of maximal entropy with disjoint support.

Levels of Generalized Expansiveness

We study a class of generalized expansive dynamical systems for which at most countable orbits can be accompanied by an arbitrary given orbit. Examples of different levels of generalized expansiveness are constructed. When the dynamical system considered has countable cardinality, several characterizations of generalized expansive homeomorphism are provided, and as a consequence examples of dynamical systems with van der Waerden depth equal to any given countable ordinal number are demonstrated, which solves open questions existing in the literature.

On the uniqueness of Gibbs states in some dynamical systems

By applying Grothendieck theory and Ruelle thermodynamic formalism, we prove that, for expansive dynamical systems and interaction potentials satisfying certain conditions of analyticity, the associated Gibbs states are unique. This allows us to draw an analogy between some quantities in classical thermodynamics and abstract dynamics in the spirit of the previous work of the authors [13].

Lyapunov spectra for KMS states on Cuntz-Krieger algebras

We study relations between $(H,\beta)$--KMS states on Cuntz--Krieger algebras and the dual of the Perron--Frobenius operator $\mathcal{L}_{-\beta H}^{*}$. Generalising the well--studied purely hyperbolic situation, we obtain under mild conditions that for an expansive dynamical system there is a one--one correspondence between $(H,\beta)$--KMS states and eigenmeasures of $\mathcal{L}_{-\beta H}^{*}$ for the eigenvalue 1. We then consider representations of Cuntz--Krieger algebras which are induced by Markov fibred systems, and show that if the associated incidence matrix is irreducible then these are $\ast$--isomorphic to the given Cuntz--Krieger algebra. Finally, we apply these general results to study multifractal decompositions of limit sets of essentially free Kleinian groups $G$ which may have parabolic elements. We show that for the Cuntz--Krieger algebra arising from $G$ there exists an analytic family of KMS states induced by the Lyapunov spectrum of the analogue of the Bowen--Series map associated with $G$. Furthermore, we obtain a formula for the Hausdorff dimensions of the restrictions of these KMS states to the set of continuous functions on the limit set of $G$. If $G$ has no parabolic elements, then this formula can be interpreted as the singularity spectrum of the measure of maximal entropy associated with $G$.

Embeddings into Symbolic Dynamical Systems

In this paper we characterize those dynamical systems ( X, T ), where X is a topological space and T is a continuous self-map on X , which are weakly embeddable into a symbolic dynamical system. As a corollary we obtain a result of Akashi regarding the embeddability of certain positively expansive dynamical systems into symbolic dynamical systems.

Positively expansive dynamical systems

We introduce the notions of weakly and strongly positively expansive (wPE and sPE, respectively) discrete dynamical systems. Both are topological generalizations of the well-known metric notion of positive expansiveness (PE). We prove that the three notions are identical on compact metrizable spaces, but not on noncompact spaces. We investigate properties of PE, wPE, and sPE dynamical systems and show how they are related. Finally, we show that the possible dynamics for wPE and sPE homeomorphisms are severely limited, and we classify sPE homeomorphisms.