Search

We introduce and study the BS dimension of subsets for free semigroup actions. In fact, we estabilish the variational principles for BS dimension and measure theoretic pressure. [ABSTRACT FROM AUTHOR]

In this paper we study topological aspects of the dynamics of the foliated horocycle flow on flat projective bundles over hyperbolic surfaces and we derive ergodic consequences. If ρ : Γ → PSL (n + 1 , R) is a representation of a non-elementary Fuchsian group Γ , the unit tangent bundle Y associated to the flat projective bundle defined by ρ admits a natural action of the affine group B obtained by combining the foliated geodesic and horocycle flows. If the image ρ (Γ) satisfies Conze-Guivarc’h conditions, namely strong irreducibility and proximality, the dynamics of the B-action is captured by the proximal dynamics of ρ (Γ) on R P n (Theorem A). In fact, the dynamics of the foliated horocycle flow on the unique B-minimal subset of Y can be described in terms of dynamics of the horocycle flow on the non-wandering set in the unit tangent bundle X of the surface S = Γ \ H (Theorem B). Assuming the existence of a continuous limit map, we prove that the B-minimal set is an attractor for the foliated horocycle flow restricted to the proximal part of the non-wandering set in Y (Theorem C). As a corollary, we deduce that the restricted flow admits a unique conservative ergodic U-invariant Radon measure (defined up to a multiplicative constant) if and only if Γ is convex-cocompact. For example, the foliated horocycle flow on the sphere bundle defined by the Cannon-Thurston map is uniquely ergodic. [ABSTRACT FROM AUTHOR]

In this paper, the entropy of discrete Heisenberg group actions is considered. Let α be a discrete Heisenberg group action on a compact metric space X. Two types of entropies, h ~ (α) and h(α) are introduced, in which h ~ (α) is defined in Ruelle's way and h(α) is defined via the natural extension of α. It is shown that when X is the torus and α is induced by integer matrices then h ~ (α) is zero and h(α) can be expressed via the eigenvalues of the matrices. [ABSTRACT FROM AUTHOR]

This paper introduces the concepts of correlation entropy and local correlation entropy for free semigroup actions on compact metric space, and investigates their underlying properties. Thereafter, we extend certain classical findings on correlation entropy and local correlation entropy to the realm of free semigroup actions. Finally, we establish the interconnections between topological entropy, measure-theoretic entropy, correlation entropy, and local correlation entropy for free semigroup actions under different conditions. [ABSTRACT FROM AUTHOR]

This paper studies topological entropy and pseudo-entropy of iterated set-valued function systems. Firstly, the notions of topological entropy defined by separating and spanning sets and by open covers are introduced respectively, and they are proved equivalent, then a formula is obtained for the topological entropy of an iterated set-valued function system concerning the corresponding skew product system, and topological entropy of iterated set-valued function systems is a topological conjugacy invariant. Finally, the notions of pseudo-entropy of set-valued function systems and iterated set-valued function systems are introduced and it is proved that the pseudo-entropy is equal to the topological entropy of iterated set-valued function systems. [ABSTRACT FROM AUTHOR]

A classical result of Fathi and Herman from 1977 states that a smooth compact connected manifold without boundary admitting a locally free action of a 1-torus, respectively, an almost free action of a 2-torus, admits a minimal diffeomorphism, respectively, a minimal flow. In the first part of our paper we study the existence of locally free and almost free actions of tori on homogeneous spaces of compact connected Lie groups, thus providing new examples of spaces admitting minimal diffeomorphisms or flows. In the second part we combine the ideas of Fathi and Herman with our recent ideas to study the existence of minimal skew products over certain minimal flows with general connected Lie groups as acting groups. Our results apply to so called flows with free cycles. In the last part of our work we study the existence of free cycles in homogeneous flows. [ABSTRACT FROM AUTHOR]

Let G be an abelian semigroup of matrices on K n ( K = C or R ). We show that if G is hypercyclic, then it has no non-zero finite orbit. This result fails if we drop the assumption that G is abelian. As a consequence, if G is abelian, it is not chaotic. On the other hand, we show that G is not minimal for n ≥ 3 , but it can be minimal for n = 1 ; for K = R , the critical number is n = 2 . [ABSTRACT FROM AUTHOR]

We consider contact foliations: objects which generalise to higher dimensions the flow of the Reeb vector field on contact manifolds. We list several properties of such foliations and propose two conjectures about the topological types of their leaves, both of which coincide with the classical Weinstein conjecture in the case of contact flows. We give positive partial results for our conjectures in particular cases—when the holonomy of the contact foliation preserves a Riemannian metric, for instance—extending already established results in the field of Contact Dynamics. [ABSTRACT FROM AUTHOR]

We show that every quasiconformal contact foliation supports an invariant metric and characterise these foliations by the dynamical property of C 1 -equicontinuity. We prove that a generalisation of the Weinstein conjecture holds for quasiconformal contact foliations, and provide a lower bound to the number of closed leaves. In particular, we show that the Weinstein conjecture holds for quasiconformal Reeb fields. [ABSTRACT FROM AUTHOR]

We prove global results about actions of cocompact lattices in higher-rank simple Lie groups on closed manifolds endowed with either a projective class of connections or a conformal class of pseudo-Riemannian metrics of signature (p, q), with min (p , q) ≥ 2 . Building on a recent article [17], provided that such a structure is locally equivalent to its model X , the main question treated here is the completeness of the associated (G , X) -structure. Because of the similarities between the model spaces of projective geometry and non-Lorentzian conformal geometry, a number of arguments apply in both contexts. We therefore present the proofs in parallel. The conclusion is that in both cases, when the real rank is maximal, the manifold is globally equivalent to either the model space X or its double cover. [ABSTRACT FROM AUTHOR]

In this paper, the shadowing property of 1-dimensional subsystems for suspension of ℤ k -actions is investigated. The concepts of pseudo orbit and shadowing property of 1-dimensional subsystems for suspension of ℤ k -actions are introduced. It is shown that the shadowing properties of these subsystems and their induced nonautonomous dynamical systems are equivalent. For the suspension ΦT of a smooth ℤ k -action T, we show that ΦT has the shadowing property along any Anosov subspace. As an application, we show that ΦT is structurally stable along any Anosov subspace. [ABSTRACT FROM AUTHOR]

Inspired to the work of Ma and Wu (Discrete Contin Dyn Syst 31:545–557, 2011) and Climenhaga (Ergodic Theory Dyn Syst 31(4):1163–1182, 2011), we introduce the new notions of topological pressure and upper (lower) capacity topological pressure of a free semigroup action of maps for finite potential functions by using the Carathéodory–Pesin structure (C-P structure) with respect to non-compact sets in this paper. Meanwhile, we also give some properties of these notions. Moreover, by Bowen's equation, we characterize the Hausdorff dimension of certain sets, whose points have the positive lower Lyapunov exponents and satisfy a so-called tempered contraction condition. [ABSTRACT FROM AUTHOR]

The invariant Borel probability measures play a crucial role in the study of the notion of the support of topological dynamical systems. Unfortunately, general systems with (spatial) discontinuity may not admit any invariant Borel probability measure. In this paper, we study the notion of the support of skew-product amenable semigroup action systems which may admit discontinuity. We also establish a characterization of strong proximality for these actions based on Banach proximality. [ABSTRACT FROM AUTHOR]

In this paper, we study metric mean dimension with potential and set up a variational principle for it, which connects rate distortion function to metric mean dimension with potential. Moreover, we give alternative definitions of metric mean dimensions with potential and show that classic metric mean dimension with potential is equal to Bowen upper and lower capacity metric mean dimensions with potential. [ABSTRACT FROM AUTHOR]

In this paper, for skew-product actions (SPAs) of amenable semigroups (and commutative semigroups) with discontinuity from the point of view of topology, we establish the Bogolyubov–Krylov theorem for the existence of invariant Borel probability measures. In particular, we obtain uniform and semi-uniform ergodic theorems for SPAs of amenable semigroups. [ABSTRACT FROM AUTHOR]

In this paper, we study chain mixing and various multi-transitivities in iterated function systems. We show that an iterated function system IFS(F ) is chain mixing if and only if IFS( F r ) × IFS( F s ) is chain transitive for some positive integers r and s. We introduce and study the notions of multi-transitive, Δ -transitive and Δ -mixing for iterated function systems. Relations are established between multi-transitivity and shadowing properties. For example, D-multi-transitivity, Δ -mixing and chain mixing are equivalent mutually for an IFS(F ) which has the shadowing property. Moreover, we point out that multi-transitivity (resp. Δ -transitivity, Δ -mixing) is different from Z-multi-transitivity (resp. Z- Δ -transitivity, Z- Δ -mixing) which is defined by Tiaoying Zeng. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the relation between topological pressure of free semigroup actions for non-compact sets proposed by Xiao and Ma (J Dynam Differ Equ, 2021) and measure-theoretic pressure of Borel probability measure. For any Borel probability measure, this paper defines lower and upper measure-theoretic pressures. Moreover, we give a lower and an upper estimations of the topological pressure of free semigroup actions by local measure-theoretic pressure. In addition, we also show that the topological pressure on a non-empty compact subset K defined in Xiao and Ma (J Dynam Differ Equ, 2021) equals to the supremum of the lower measure-theoretic pressure taken over all probability measures supported on K. [ABSTRACT FROM AUTHOR]

We improve a recent construction of Andrés Navas to produce the first examples of $C^2$ -undistorted diffeomorphisms of the interval that are $C^{1+\alpha }$ -distorted (for every ${\alpha). We do this via explicit computations due to the failure of an extension to class $C^{1+\alpha }$ of a classical lemma related to the work of Nancy Kopell. [ABSTRACT FROM AUTHOR]

In this paper we prove a perturbative result for a class of ${\mathbb Z}^2$ actions on Heisenberg nilmanifolds that have Diophantine properties. Along the way we prove cohomological rigidity and obtain a tame splitting for the cohomology with coefficients in smooth vector fields for such actions. [ABSTRACT FROM AUTHOR]

In this paper, we study the centralizer of a partially hyperbolic diffeomorphism on ${\mathbb T}^3$ which is homotopic to an Anosov automorphism, and we show that either its centralizer is virtually trivial or such diffeomorphism is smoothly conjugate to its linear part. [ABSTRACT FROM AUTHOR]

Let F and K be commuting C ∞ diffeomorphisms of the cylinder T × R that are, respectively, close to F 0 (x , y) = (x + ω (y) , y) and T α (x , y) = (x + α , y) , where ω (y) is non-degenerate and α is Diophantine. Using the KAM iterative scheme for the group action we show that F and K are simultaneously C ∞ -linearizable if F has the intersection property (including the exact symplectic maps) and K satisfies a semi-conjugacy condition. We also provide examples showing necessity of these conditions. As a consequence, we get local rigidity of certain class of Z 2 -actions on the cylinder, generated by commuting twist maps. [ABSTRACT FROM AUTHOR]

In this paper we study Zimmer's conjecture for $C^{1}$ actions of lattice subgroup of a higher-rank simple Lie group with finite center on compact manifolds. We show that when the rank of an uniform lattice is larger than the dimension of the manifold, then the action factors through a finite group. For lattices in ${\rm SL}(n, {{\mathbb {R}}})$ , the dimensional bound is sharp. [ABSTRACT FROM AUTHOR]

In this survey we will present the symbolic extension theory in topological dynamics, which was built over the past twenty years. [ABSTRACT FROM AUTHOR]

The aim of this manuscript is to study some local properties of the topological entropy of a free semigroup action. In order to do that we focus on the set of entropy points of a free semigroup action, show that this set carries the full entropy of the system (which, with respect to the chaocity of the system, gives a fundamental relevance to such set). We prove that the topological entropy of a free semigroup action focuses on a countable set of points. Our results are inspired by the ones presented in [1]. [ABSTRACT FROM AUTHOR]

We consider continuous free semigroup actions generated by a family $(g_y)_{y \,\in \, Y}$ of continuous endomorphisms of a compact metric space $(X,d)$ , subject to a random walk $\mathbb P_\nu =\nu ^{\mathbb N}$ defined on a shift space $Y^{\mathbb N}$ , where $(Y, d_Y)$ is a compact metric space with finite upper box dimension and $\nu $ is a Borel probability measure on Y. With the aim of elucidating the impact of the random walk on the metric mean dimension, we prove a variational principle which relates the metric mean dimension of the semigroup action with the corresponding notions for the associated skew product and the shift map $\sigma $ on $Y^{\mathbb {N}}$ , and compare them with the upper box dimension of Y. In particular, we obtain exact formulas whenever $\nu $ is homogeneous and has full support. We also discuss several examples to enlighten the roles of the homogeneity, of the support and of the upper box dimension of the measure $\nu $ , and to test the scope of our results. [ABSTRACT FROM AUTHOR]

In this article, we showed that positive relative entropy implies mean Li-Yorke chaos along some good sequences in the fiber. Our result extended Li and Qiao's result in [Monatsh. Math., 186 (2018), 153–173]. [ABSTRACT FROM AUTHOR]