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We introduce and explore new notions of pressure of multipotentials Φ = (φ j) j ∈ Ω along arbitrary sets of trajectories Y ⊂ Σ Ω + × X for semigroups generated by a set G 1 = (f j) j ∈ Ω of continuous maps on a compact metric space X. The most useful among them is the amalgamated pressure P A (Φ , Y , G 1) . Amalgamated pressure is applied to formulas or estimates for Hausdorff dimension. First if G 1 is a finite set of conformal maps on a metric space X, we obtain a formula for HD(Y) for sets Y on which the G 1 -Lyapunov exponents are positive. In particular, this gives a formula for H D (J f ∩ J g) if J f , J g are conformal repellers in R k with non-empty intersection. This applies also to semigroups generated by parabolic maps. We obtain the dimension of the projection set G μ of the set G μ of generic trajectories for a G 1 -expanding measure μ on Σ Ω + × X . Next we study the case when G 1 is a finite set of C 2 maps on a manifold M and there exist G 1 -partial stable and unstable cone fields on a compact set Λ . For a non-uniformly hyperbolic subset Y ⊂ Λ we apply the amalgamated pressure of the unstable multipotential Φ u , to estimate the Hausdorff dimension of the slices through Y with arbitrary submanifolds transversal to stable cones. We introduce then a measure-theoretic amalgamated entropy h A (μ , G 1) for the marginal measure of any F-invariant ergodic measure μ on Σ Ω + × X , and prove a partial Variational Principle for the amalgamated pressure. Semigroups of toral endomorphisms are studied also, and computations are provided. We obtain also a new notion of pressure for random dynamical systems and random potentials. [ABSTRACT FROM AUTHOR]

This article studies past attractors and future repellers of topological cocycles conducted by a group G satisfying Ore's condition. The limit behaviour depends on the Green $ \mathcal {L} $ L -pre-order on a right reversible semigroup S of G. An existence and uniqueness theorem for past attractors is proved. A topological method of extending functions is used in order to describe the past attractors by means of skew-product prolongations in the extended space. Illustrative examples of past attractor and future repeller are provided. [ABSTRACT FROM AUTHOR]

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 define and study the topological version of the specification property, called open cover specification property. We explore its connections with different types of mixing properties, such as topological mixing, total transitivity, weak mixing, and the non-wandering property. We also establish that, in the presence of positive orbit expansivity on a compact Hausdorff space, the open cover specification property implies local eventual onto behaviour. Additionally, we calculate the topological entropy of a dynamical system on a compact Hausdorff space, showing that it can be expressed in terms of periodic points when the map satisfies both the open cover specification property and the positive orbit expansivity. [ABSTRACT FROM AUTHOR]

Based on a fixed point theorem by Shi and Ye for amenable group actions on dendrites, we show that if X is a Peano continuum having the property that for any ϵ > 0 there are only finitely many circles in X with diameters ≥ ϵ , then there exists no expansive polycyclic group action on X. As a corollary, we obtain that local dendrites and Peano continua containing no θ -curves admit no expansive polycyclic group actions. [ABSTRACT FROM AUTHOR]

Pavlov [ Adv. Math. 295 (2016), 250–270; Nonlinearity 32 (2019), 2441–2466] studied the measures of maximal entropy for dynamical systems with weak versions of specification property and found the existence of intrinsic ergodicity would be influenced by the assumptions of the gap functions. Inspired by these, in this article, we study the dynamical systems with non-uniform specification property. We give some basic properties these systems have and give an assumption for the gap functions to ensure the systems have the following five properties: CO-measures are dense in invariant measures; for every non-empty compact connected subset of invariant measures, its saturated set is dense in the total space; ergodic measures are residual in invariant measures; ergodic measures are connected; and entropy-dense. In addition, we will give examples to show the assumption is optimal. [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]

Let π: (X, T) → (Y, S) be a factor map between two topological dynamical systems, and F a Furstenberg family of ℤ. We introduce the notion of relative broken F -sensitivity. Let F s (resp. F pubd , F inf ) be the families consisting of all syndetic subsets (resp. positive upper Banach density subsets, infinite subsets). We show that for a factor map π: (X, T) → (Y, S) between transitive systems, π is relatively broken F -sensitive for F = F s or F pubd if and only if there exists a relative sensitive pair which is an F -recurrent point of (Rπ, T(2)); is relatively broken F inf -sensitive if and only if there exists a relative sensitive pair which is not asymptotic. For a factor map π: (X, T) → (Y, S) between minimal systems, we get the structure of relative broken F -sensitivity by the factor map to its maximal equicontinuous factor. [ABSTRACT FROM AUTHOR]

The weak mean equicontinuity for a countable discrete amenable group G acting continuously on a compact metrizable space X is studied. It is shown that weak mean equicontinuity, continuously pointwise ergodicity and uniformity are coincided. Moreover, we prove that (X, G) is mean equicontinuous if and only if the product system (X × X , G) is weak mean equicontinuous. [ABSTRACT FROM AUTHOR]

For a topological dynamical system (X, T), l ∈ N and x ∈ X , let N l (X) and L x l (X) be the orbit closures of the diagonal point (x , ... , x) (l times) under the actions G l and τ l respectively, where G l is generated by T × ... × T (l times) and τ l = T × ... × T l . In this paper, we show that for a minimal system (X, T) and d , l ∈ N , the maximal d-step pro-nilfactor of (N l (X) , G l) is (N l (X d) , G l) , where X d is the d-step pronilfactor of (X, T). Meanwhile, when (X, T) is a minimal nilsystem, we also calculate the pro-nilfactors of the system (L x l (X) , τ l) for almost every x w.r.t. the Haar measure. In particular, there exists a minimal 2-step nilsystem (Y, T) and a countable subset Ω of Y such that for every y ∈ Y \ Ω the maximal equicontinuous factor of (L y 2 (Y) , τ 2) is not (L π 1 (y) 2 (Y 1) , τ 2) , where Y 1 is the maximal equicontinuous factor of (Y, T) and π 1 : Y → Y 1 is the factor map. [ABSTRACT FROM AUTHOR]

We study general random dynamical systems of continuous maps on some compact metric space. Assuming a local contraction condition and proximality, we establish probabilistic limit laws such as the (functional) central limit theorem, the strong law of large numbers, and the law of the iterated logarithm. Moreover, we study exponential synchronization and synchronization on average. In the particular case of iterated function systems on S 1 , we analyze synchronization rates and describe their large deviations. In the case of C 1 + β -diffeomorphisms, these deviations on random orbits are obtained from the large deviations of the expected Lyapunov exponent. [ABSTRACT FROM AUTHOR]

We establish sufficient and necessary conditions for the joint transitivity of linear iterates in a minimal topological dynamical system with commuting transformations. This result provides the first topological analogue of the classical Berend and Bergelson joint ergodicity criterion in measure-preserving systems.

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 $ (X, f) $ (X , f) be a dynamical system, i.e. X is a compact metric space and f is a continuous self-map on X and let $ K(X) $ K (X) , $ M(X) $ M (X) and $ \mathbb {F}(X) $ F (X) be the sets of all non-empty compact subsets of X, Borel probability measures on X and upper semi-continuous fuzzy sets on X, respectively. Then $ K(X) $ K (X) , $ M(X) $ M (X) and $ \mathbb {F}(X) $ F (X) are metric spaces under Hausdorff metric, prohorov metric and level-wise metric, respectively. Therefore, $ (X, f) $ (X , f) naturally induces three new systems $ (K(X), \bar {f}) $ (K (X) , f ¯) , $ (M(X), \hat {f}) $ (M (X) , f ^) and $ (\mathbb {F}(X), \tilde {f}) $ (F (X) , f ~). In this article, we investigate the connection of $ (r,s) $ (r , s) -sensitivity, $ (r,s) $ (r , s) -asymptotic sensitivity, $ (r,s) $ (r , s) -Li-Yorke sensitivity and Δ-transitivity of $ (X, f) $ (X , f) and its induced systems $ (K(X),\bar {f}) $ (K (X) , f ¯) , $ (M(X),\hat {f}) $ (M (X) , f ^) and $ (\mathbb {F}(X),\tilde {f}) $ (F (X) , f ~) and we obtain some desired results. For instance, we prove that $ (K(X),\bar {f}) $ (K (X) , f ¯) is $ (r,s) $ (r , s) -sensitive ⇔ $ (\mathbb {F}^{1}(X),\widetilde {f_{g}}) $ (F 1 (X) , f g ~) is $ (r,s) $ (r , s) -sensitive for each $ g\in D_{m}(I) $ g ∈ D m (I) satisfying $ g^{-1}(1)=\{1\} $ g − 1 (1) = { 1 } ; $ (X,f) $ (X , f) is Δ-transitive ⇔ $ (K(X),\bar {f}) $ (K (X) , f ¯) is Δ-transitive ⇔ $ (M(X),\hat {f}) $ (M (X) , f ^) is Δ-transitive $ \Leftrightarrow \,(\mathbb {F}^{1}(X),\tilde {f}) $ ⇔ (F 1 (X) , f ~) is Δ-transitive. [ABSTRACT FROM AUTHOR]

We introduce notions of continuous orbit equivalence and its one-sided version for countable left Ore semigroup actions on compact spaces by surjective local homeomorphisms, and characterize them in terms of the corresponding transformation groupoids and their operator algebras. In particular, we show that two essentially free semigroup actions on totally disconnected compact spaces are continuously orbit equivalent if and only if there is a canonical abelian subalgebra preserving C*-isomorphism between the associated transformation groupoid C*-algebras. We also give some examples of orbit equivalence, consider the special case of semigroup actions by homeomorphisms and relate continuous orbit equivalence of semigroup actions to that of the associated group actions. [ABSTRACT FROM AUTHOR]

In this paper, we study the centralizer of a separating continuous flow without fixed points. We show that if M is a compact metric space and $ \phi _t:M\to M $ ϕ t : M → M is a separating flow without fixed points, then $ \phi _t $ ϕ t has a quasi-trivial centralizer, that is, if a continuous flow $ \psi _t $ ψ t commutes with $ \phi _t $ ϕ t , then there exists a continuous function $ A: M\to \mathbb {R} $ A : M → R which is invariant along the orbit of $ \phi _t $ ϕ t such that $ \psi _t(x)=\phi _{A(x)t}(x) $ ψ t (x) = ϕ A (x) t (x) holds for all $ x\in M $ x ∈ M. We also show that if M is a compact Riemannian manifold without boundary and $ \Phi _u $ Φ u is a homogenous separating $ C^1 $ C 1 $ \mathbb {R}^m $ R m -action on M, then $ \Phi _u $ Φ u has a quasi-trivial centralizer, that is, if $ \Psi _u $ Ψ u is a $ \mathbb {R}^{ m} $ R m -action on M commuting with $ \Phi _u $ Φ u , then there is a continuous map $ A: M\to \mathcal {M}_{m\times m}(\mathbb {R}) $ A : M → M m × m (R) which is invariant along orbit of $ \Phi _u $ Φ u such that $ \Psi _{u}(x)=\Phi _{A(x)u}(x) $ Ψ u (x) = Φ A (x) u (x) for all $ x\in M $ x ∈ M. These improve Theorem 1 of [M. Oka, Expansive flows and their centralizers, Nagoya Math. J. 64 (1976), pp. 1–15.] and Theorem 2 of [W. Bonomo, J. Rocha, and P. Varandas, The centralizer of Komuro-expansive flows and expansive $ \mathbb {R}^d $ R d -actions, Math. Z. 289(3–4) (2018), pp. 1059–1088.] respectively. [ABSTRACT FROM AUTHOR]

We compare various notions of proper discontinuity for group actions. We alsodiscuss fundamental domains and criteria for cocompactness.

We give a simple proof that any dendrite map for which the set of endpoints is closed and countable fulfilled Sarnak Möbius disjointness. We further notice that the Smital-Ruelle property can be extended to the class of dendrites with closed and countable endpoints. Our proof use only Dirichlet theorem. [ABSTRACT FROM AUTHOR]

Let X be a local dendrite and let $ f:X\to X $ f : X → X be a continuous map. Denote by $ P(f) $ P (f) and $ R(f) $ R (f) the sets of periodic and recurrent points of f, respectively. We say that f is relatively pointwise recurrent if $ \overline {R(f)}=X $ R (f) ¯ = X. We show that, if X is almost meshed (i.e the union of the interior of all free arcs is dense in X), then f is relatively pointwise recurrent if and only if either f is topologically conjugate to an irrational rotation on the circle or it has a dense set of periodic points. [ABSTRACT FROM AUTHOR]

We recall various notions of size for a topological semigroup T, not necessarily discrete, such as GH-syndetic set, syndetic set, and positive Følner density set. We give new information about these sets and give some examples to study the relation between them. Let φ : T × X → X , or simply (T, X), be any dynamical system on a space X with a topological semigroup T. We say that x ∈ X is a uniformly recurrent point, almost periodic point of von Neumann, or weakly uniformly recurrent point, if the return time set N(x, U) is syndetic, GH-syndetic, or d F ϕ (N (x , U) > 0 , respectively, where U is a neighborhood of the point x and N (x , U) = { t : t x ∈ U } . It is known that there is no relation between the set of uniformly recurrent points and the set of almost periodic points of von Neumann for a semiflow (T, X). We give further examples for it. We introduce a notion of f-uniformly recurrent point, where f : X → R + is upper semicontinuous and show that x ∈ X is uniformly recurrent if and only if it is an f-uniformly recurrent point for every upper semicontinuous f : X → R + on the regular space X. In the case of metric space X, f : X → R + is a continuous function. Also, x ∈ X is a uniformly recurrent point if and only if x ∈ Ax ¯ for every thick set A of T. Assume that S is a closed normal non-trivial subsemigroup of T. We prove that every uniformly recurrent point of (S, X) is a uniformly recurrent point of (T, X). The converse holds if T is a discrete semigroup. Let (T, X) be a semiflow on topological space X. Then we show that every two nonempty open sets in X share an orbit of a weakly uniformly recurrent point of (T, X) if and only if (T, X) is a topologically transitive with a dense set of weakly uniformly recurrent points. Finally, we give topological version of sensitive dependence on the initial condition for semiflow (T, X) on topological space X and we show that if the semiflow (T, X) is nonminimal and every two non-empty open sets share an orbit of a weakly uniformly recurrent point, then (T, X) is syndetic-sensitive. [ABSTRACT FROM AUTHOR]

The preimage entropy provides a quantitative estimate of how “invertible” a system is. Once there are several examples where the preimage entropy is infinite, it cannot provide more information. Thus, we introduce the concept of preimage metric mean dimension, and study many properties of it. Meanwhile, we provide many examples to compute their preimage mean metric dimension. [ABSTRACT FROM AUTHOR]

Given a countable group G and a G -flow X , a probability measure $\mu $ on X is called characteristic if it is $\mathrm {Aut}(X, G)$ -invariant. Frisch and Tamuz asked about the existence of a minimal G -flow, for any group G , which does not admit a characteristic measure. We construct for every countable group G such a minimal flow. Along the way, we are motivated to consider a family of questions we refer to as minimal subdynamics: Given a countable group G and a collection of infinite subgroups $\{\Delta _i: i\in I\}$ , when is there a faithful G -flow for which every $\Delta _i$ acts minimally? [ABSTRACT FROM AUTHOR]

In this paper we consider the question of what abelian groups can arise as the K-theory of C ∗ -algebras arising from minimal dynamical systems. We completely characterize the K-theory of the crossed product of a space X with finitely generated K-theory by an action of the integers and show that crossed products by a minimal homeomorphisms exhaust the range of these possible K-theories. Moreover, we may arrange that the minimal systems involved are uniquely ergodic, so that their C ∗ -algebras are classified by their Elliott invariants. We also investigate the K-theory and the Elliott invariants of orbit-breaking algebras. We show that given arbitrary countable abelian groups G 0 and G 1 and any Choquet simplex Δ with finitely many extreme points, we can find a minimal orbit-breaking relation such that the associated C ∗ -algebra has K-theory given by this pair of groups and tracial state space affinely homeomorphic to Δ . We also improve on the second author's previous results by using our orbit-breaking construction to C ∗ -algebras of minimal amenable equivalence relations with real rank zero that allow torsion in both K 0 and K 1 . These results have important applications to the Elliott classification program for C ∗ -algebras. In particular, we make a step towards determining the range of the Elliott invariant of the C ∗ -algebras associated to étale equivalence relations. [ABSTRACT FROM AUTHOR]

Let (X, G) be a G-action topological system, where G is a countable infinite discrete amenable group and X a compact metric space. We prove a variational principle for topological entropy of saturated sets for systems which have the specification property and uniform separation property. We show that certain algebraic actions satisfy these two conditions. We give an application in multifractal analysis. [ABSTRACT FROM AUTHOR]

We give a new categorical approach to the Halmos–von Neumann theorem for actions of general topological groups. As a first step, we establish that the categories of topological and measure-preserving irreducible systems with discrete spectrum are equivalent. This allows to prove the Halmos–von Neumann theorem in the framework of topological dynamics. We then use the Pontryagin and Tannaka–Krein duality theories to obtain classification results for topological and then measure-preserving systems with discrete spectrum. As a byproduct, we obtain a complete isomorphism invariant for compactifications of a fixed topological group. [ABSTRACT FROM AUTHOR]

Relationships between a chaotic behaviour and closely related properties of topological transitivity, sensitivity to initial conditions, density of closed orbits of homeomorphism groups and their countable products are investigated. We provide a large number of new examples of chaotic groups of homeomorphisms of countable products of various metrizable topological spaces, including infinite-dimensional topological manifolds, whose factors can be as noncompact surfaces, so triangulable closed manifolds of an arbitrary dimension. [ABSTRACT FROM AUTHOR]