In this paper, we study the weak mean metric and give some properties by replacing the Besicovitch pseudometric with weak mean metric in the definition of mean equicontinuity and mean sensitivity. We study an opposite side of weak mean equicontinuity, namely strong mean sensitivity and we obtain some dichotomies: minimal topological dynamical systems are either weakly mean equicontinuous or strongly mean sensitive and transitive topological dynamical systems are either almost weakly mean equicontinuous or strongly mean sensitive. Furthermore, motivated by the localized idea of sensitivity, we introduce some notions of new version sensitive tuples and study the properties of these sensitive tuples, we show that a transitive dynamical system is strongly mean sensitive if and only if it admits a strongly mean sensitive tuple. Finally, we introduce the notion of weak mean equicontinuity of a topological dynamical system with respect to a given continuous function f, and we show that a topological dynamical system is weakly mean equicontinuous then it is weakly mean equicontinuous with respect to every continuous function.

In 2022, Li, Ye and Yu introduced multivariate sensitivity version of notions of mean m-Sensitivity and m-sensitivity in the mean for m ≥ 2. In this manuscript, we mainly focus on the investigation of multivariate sensitivity in mean forms. First, we prove that for a linear system, the equivalence between mean m-sensitivity and m-sensitivity in the mean holds without any more conditions. Subsequently, we introduce the notion of m-equicontinuity in the mean, and obtain an Auslander–Yorke type dichotomy between m-equicontinuity in the mean and m-sensitivity in the mean for minimal systems. As a consequence, we demonstrate that the equivalence between mean m-sensitivity and m-sensitivity in the mean is valid under the condition of minimality for a general system, thereby affirmatively resolving the conjecture proposed in [Li, J., Ye, X. D., Yu, T.: Equicontinuity and sensitivity in mean forms. J. Dynam. Differential Equations, 34, 133–154 (2022)].

We study expansive actions of groups (not necessarily countable) on compact Hausdorff spaces (not necessarily metrizable). We define a series of topological Markov properties in terms of uniformities. For amenable groups, we prove that an expansive dynamical system with positive topological entropy and the strong topological Markov property admits an off-diagonal homoclinic pair. On the one hand, this result remains valid if the strong topological Markov property is replaced by the shadowing property. On the other hand, we deduce that a strongly irreducible and splicable subshift admits two distinct, almost equal configurations. Additionally, we show that a minimal expansive dynamical system admits no off-diagonal homoclinic pairs if and only if it satisfies the topological Markov property. (c) 2026 Elsevier Masson SAS. All rights are reserved, including those for text and data mining, AI training, and similar technologies.

In this paper, we investigate the Weyl mean orbital pseudo-metric for Polish dynamical systems and its connections to the properties of the space of invariant measures. We establish equivalent conditions under which the set of invariant measures generated by periodic points is dense in the set of ergodic measures, thereby providing a deeper understanding of the relationship between periodic behavior and measure-theoretic properties in the field of Weyl mean orbital pseudo-metric.

Building on the work of Dou, Fan and Qiu, who introduced Bowen entropy for flows via reparametrization ball techniques and established a variational principle for fixed-point free flows, this paper further develops the variational principle of Bowen entropy in such systems. We introduce two new notions of measure-theoretic entropy tailored to fixed-point free flows and establish both a variational principle and an inverse variational principle relating these entropies to Bowen entropy. Finally, we give some calculations about Bowen entropy.

We study various types of mean equicontinuity and mean sensitivity for actions of countable discrete amenable groups. First, we show that ℱ -mean equicontinuity and ℱ -equicontinuity in the mean are equivalent when ℱ is a two-sided Følner sequence for a countable discrete amenable group. We then prove that, for ℱ -mean equicontinuity and ℱ -mean sensitivity, the Hausdorff and uniform versions are equivalent on compact Hausdorff spaces and to the classical definitions on compact metrizable spaces. Finally, for point-transitive dynamical systems, we establish a dichotomy theorem between uniform ℱ -mean equicontinuity and uniform ℱ -mean sensitivity.

In this paper, we investigate removability for generalized John metric spaces. We prove that X is a generalized John metric space if and only if X\ P is a generalized John metric space, where P is a countable subset of X which satisfies a quasihyperbolic b-separation condition with parameter b>0 .

In this paper we introduce a new notion of topological entropy for non-autonomous iterated function systems, which differs from the definition proposed by Ghane and Nazarian Sarkooh (2019) [12]. We obtain a Bowen-type inequality under factor maps and establish a partial variational principle relating measure-theoretic entropy and topological entropy. This strengthens the partial variational principle proved by Ju et al. (2024) [18]. Finally, we study the local entropy of non-autonomous iterated function systems and use it to give another proof of the partial variational principle. (c) 2026 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.

In 2025, Cheng introduced the notion of extended receptive entropy for two independent actions, generalizing the receptive topological entropy developed in Ghys et al. [Acta Math. 160(1-2), 105–142 (1988)] and Biś et al. [Qual. Theory Dyn. Syst. 20(2), paper no. 50 (2011)]. We define the extended Pesin receptive entropy of subsets defined by the Carathéodory–Pesin structure for two independent actions, and compare it with Cheng’s extended receptive entropy. We further establish a variational principle and an inverse variational principle for extended Pesin receptive entropy of subsets. Finally, we prove a Billingsley type theorem for extended Pesin receptive entropy.

In this manuscript, we mainly investigate the topological pressure for iterated function systems on a compact metric space defined by Wang and Liao [Dynam. Syst., 2021, 36(3): 483-506]. Given a factor map, we establish a formula of topological pressure of a factor map, which generalizes Bowen's inequality in [Trans. Amer. Math. Soc., 1971, 153: 401-414.] to iterated function systems. Consequently, we further study the power rule of a topological pressure for iterated function systems.

We generalize Fathi's results by showing that a compact metrizable space admits an fiber expansive homeomorphism if and only if it has a compatible hyperbolic metric. Moreover, we prove that a compact metrizable space admits an fiber expansive homeomorphism if and only if it has a generator in detail. Furthermore, we show that a fiber expansive homeomorphism has finite fiber topological entropy. Finally, we show that fiber Lyapunov exponents for a fiber expansive system are different from zero, indicating that the system presents a chaotic system. Meanwhile, we also prove that negative fiber Lyapunov exponents for compact invariant sets of a dynamical system imply that the compact set is a fiber attractor. (c) 2024 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.

This manuscript investigates polynomial entropy for systems with two independent transformations. First, we introduce the notion of polynomial extended topological entropy for two independent maps, and compare this notion with the polynomial entropy given by Marco (Regul Chaotic Dyn 18:623 655, 2013). Subsequently, using Carathéodory-Pesin structure, we develop polynomial extended entropy for subsets, generalizing Marco’s weak polynomial entropy framework (Marco in Regul Chaotic Dyn 18:623 655, 2013). Finally, we prove a variational principle and an inverse variational principle for this polynomial extended entropy.

In this article, we introduce the notion of quasi-uniform entropy under free semigroup actions on quasi-uniform spaces, which can be viewed as a quasi-uniform version generalization of the topological entropy under free semigroup actions defined by Bufetov [Topological entropy of free semigroup actions and skew-product transformations, J. Dynam. Control Syst. 5(1) (1999), pp. 137-143] and Wang et al. [On the topological entropy of free semigroup actions, J. Math. Anal. Appl. 435(2) (2016), pp. 1573-1590]. Subsequently, we establish some fundamental properties of this entropy. In addition, we show that for a totally bounded quasi-uniform space, the quasi-uniform K-entropy under free semigroup action is equal to the quasi-uniform entropy of its extension to the bicompletion.

Consider a topological dynamical system (X, T) endowed with the metric d. We introduce a novel function as BF(x, y) = lim supn-m ->+infinity inf sigma is an element of Sn,m 1 d(Tkx, T sigma(k) y), where the permutation group Sn,m is utilized. It is demonstrated that BF (x, y) exists when x, y is an element of X are uniformly generic points. Leveraging this function, we introduce the concept of weak Banach mean equicontinuity and establish that the dynamical system (X, T) exhibits weak Banach mean equicontinuity if and only if the all f is an element of C(X). Finally, we demonstrate that in the case of a transitive system, the equivalence between weak Banach mean equicontinuity and weak mean equicontinuity is established.

In this paper, we mainly focus on the upper metric mean dimension with potential and BS dimension of a factor map. We aim to build a link between the localize manifestations of upper metric mean dimension with potential (BS dimension) and the overarching upper metric mean with potential dimension (BS dimension). (c) 2025 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.

In this manuscript, we consider a non-autonomous dynamical system. Using the Carathéodory structure, we define a BS dimension on an arbitrary subset and obtain a Bowen’s equation that illustrates the relation of the BS dimension to the Pesin-Pitskel topological pressure given by Nazarian [24]. Moreover, we establish a variational principle and an inverse variational principle for the BS dimension of non-autonomous dynamical systems. Finally, we also get an analogue of Billingsley’s theorem for the BS dimension of non-autonomous dynamical systems.

In this paper, we study the relationship of the metric mean dimension between a topological dynamical system (X,T, d) and the induced hyperspace topological dynamical system (2(X), 2(T), d(H)). Specially, for the topological dynamical system with metric mean dimension is zero, we give three different examples to show that the metric mean dimension of hyperspace dynamical system is zero, finite positive number and infinity, respectively.

In this paper, we generalize the results of Fathi by establishing that a compact metrizable space admits an expansive finitely generated monoid action if, and only if, it possesses a compatible hyperbolic metric. Furthermore, we demonstrate the equivalence of the concepts of expansiveness, increasing small distances, and expanding small distances within a rather general framework. Additionally, we affirm that a compact metrizable space admits an expansive countable group action precisely when it has a generator. Moreover, we prove that the expansiveness property of group actions is inherited by finite-index subgroups and finite extensions. Lastly, we exhibit that the Lyapunov exponents for an expansive system are necessarily nonzero, thereby indicating that such a system exhibits chaotic behavior. Concurrently, we also demonstrate that negative Lyapunov exponents for compact invariant sets of a dynamical system imply that the compact set in question functions as an attractor. (c) 2025 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.

In this manuscript, we aim to investigate the topological entropy defined on compact uniform spaces. First, we introduce the notions of measure-theoretic entropy and topological entropy of subsets on compact uniform spaces. Subsequently, we obtain a variational principle for topological entropy of subsets on compact uniform spaces.