In this paper, it is shown that for a residual set of points in a totally minimal system with finitely many commuting homeomorphisms, the set of return times to any non-empty open set contains a subset with positive multidimensional multiplicative upper Banach density, extending a previous result by Glasscock, Koutsogiannis and Richter. Meanwhile, we give some combinatorial properties of the sets with positive multidimensional multiplicative upper Banach density.

In this paper, we define and study polynomial entropy on an arbitrary subset and local measure theoretic polynomial entropy for any Borel probability measure on a compact metric space, and investigate the relation between local measure-theoretic polynomial entropy of Borel probability measures and polynomial entropy on an arbitrary subset. Also, we establish a variational principle for polynomial entropy on compact subsets in the context of amenable group actions.

In this paper, forward expansiveness and entropies of ``subsystems''$^{2)}$ of $\mathbb{Z}_+^k$-actions are investigated. Let $\alpha$ be a $\mathbb{Z}_+^k$-action on a compact metric space. For each $1\le j\le k-1$, denote $\mathbb{G}_j^+=\{V_+:=V\cap\mathbb{R}_+^k: V \text{ is a } j\text{-dimensional subspace of } \mathbb{R}^k\}$. We consider the forward expansiveness and entropies for $\alpha$ along $V_+\in \mathbb{G}_j^+$. Adapting the technique of ``coding'', which was introduced by M. Boyle and D. Lind to investigate expansive subdynamics of $\mathbb{Z}^k$-actions, to the $\mathbb{Z}_+^k$ cases, we show that the set $\mathbb{E}^+_j(\alpha)$ of forward expansive $j$-dimensional $V_+$ is open in $\mathbb{G}_j^+$. The topological entropy and measure-theoretic entropy of $j$-dimensional subsystems of $\alpha$ are both continuous in $\mathbb{E}^+_j(\alpha)$, and moreover, a variational principle relating them is obtained.
For a $1$-dimensional ray $L\in\mathbb{G}_1^+$, we relate the $1$-dimensional subsystem of $\alpha$ along $L$ to an i.i.d. random transformation. Applying the techniques of random dynamical systems we investigate the entropy theory of $1$-dimensional subsystems. In particular, we propose the notion of preimage entropy (including topological and measure-theoretical versions) via the preimage structure of $\alpha$ along $L$. We show that the preimage entropy coincides with the classical entropy along any $L\in \mathbb{E}_1^+(\alpha)$ for topological and measure-theoretical versions respectively. Meanwhile, a formula relating the measure-theoretical directional preimage entropy and the folding entropy of the generators is obtained.

Let X be a regular curve and n be a positive integer such that for every nonempty open set U ⊂ X, there is a nonempty connected open set V ⊂ U with the cardinality |∂_X(V)|≤ n. We show that if X admits a sensitive action of a group G, then G contains a free subsemigroup and the action has positive geometric entropy. As a corollary, X admits no sensitive nilpotent group action.

In this article, we prove the existence of quasi-periodic solutions and the boundedness of all solutions of the $p$-Laplacian equation $(\phi_p(x'))'+a\phi_p(x^+)-b\phi_p(x^-)=g(x,t)+f(t)$, where $g(x,t)$ and $f(t)$ are quasi-periodic in $t$ with Diophantine frequency. A new method is presented to obtain the generating function to construct canonical transformation by solving a quasi-periodic homological equation.

In this paper, we study unstable topological pressure for C¹-smooth partially hyperbolic diffeomorphisms with sub-additive potentials. Moreover, without any additional assumption, we have established the expected variational principle which connects this unstable topological pressure and the unstable measure theoretic entropy, as well as the corresponding Lyapunov exponent.

In this paper, the bifurcation of limit cycles for planar piecewise smooth systems is studied which is separated by a straight line. We give a new form of Abelian integrals for piecewise smooth systems which is simpler than before. In application, for piecewise quadratic system the existence of 10 limit cycles and 12 small-amplitude limit cycles is proved respectively.

In this paper, we investigate the topological stability and pseudo-orbit tracing property for homeomorphisms on uniform spaces. We introduce the concept of topological stability for homeomorphisms on compact uniform spaces and prove that if a homeomorphism on a compact uniform space is expansive and has pseudo-orbit tracing property, then it is topologically stable. Moreover, we discuss the topological stability for homeomorphisms on uniform spaces from the view of localization. We introduce definitions of topologically stable point and shadowable point for homeomorphisms on uniform spaces and show that every shadowable point of an expansive homeomorphism on a compact uniform space is topologically stable.

We study several stronger versions of sensitivity for minimal group actions, including n-sensitivity, thick n-sensitivity and blockily thick n-sensitivity, and characterize them by the regionally proximal relation.

We study bi-Lyapunov stable homoclinic classes for a C¹ generic flow on a closed Riemannian manifold and prove that such a homoclinic class contains no singularity. This enables a parallel study of bi-Lyapunov stable dynamics for flows and for diffeomorphisms. For example, we can then show that a bi-Lyapunov stable homoclinic class for a C¹ generic flow is hyperbolic if and only if all periodic orbits in the class have the same stable index.

In this paper we introduce the notions of (Banach) density-equicontinuity and densitysensitivity. On the equicontinuity side, it is shown that a topological dynamical system is densityequicontinuous if and only if it is Banach density-equicontinuous. On the sensitivity side, we introduce the notion of density-sensitive tuple to characterize the multi-variant version of density-sensitivity. We further look into the relation of sequence entropy tuple and density-sensitive tuple both in measuretheoretical and topological setting, and it turns out that every sequence entropy tuple for some ergodic measure on an invertible dynamical system is density-sensitive for this measure; and every topological sequence entropy tuple in a dynamical system having an ergodic measure with full support is densitysensitive for this measure.

We show that for every topological dynamical system with the approximate product property, zero topological entropy is equivalent to unique ergodicity. Equivalence of minimality is also proved under a slightly stronger condition. Moreover, we show that unique ergodicity implies the approximate product property if the system has periodic points.