Related papers: Irregular sets, the $\beta$-transformation and the…

Fix a compact metric space $X$ with finite topological dimension. Let $C^{0}(X)$ be the space of continuous maps on $X$ and $ H^{\alpha}(X)$ the space of $\alpha$-H\"older continuous maps on $X$, for $\alpha\in (0,1].$ $H^{1}(X)$ is the…

Hamiltonian dynamical systems tend to have infinitely many periodic orbits. For example, for a broad class of symplectic manifolds almost all levels of a proper smooth Hamiltonian carry periodic orbits. The Hamiltonian Seifert conjecture is…

The special structures that arise in symplectic topology (particularly Gromov--Witten invariants and quantum homology) place as yet rather poorly understood restrictions on the topological properties of symplectomorphism groups. This…

The paper studies the structure of $\omega$-limit sets of map $\tilde{f}$ induced on the hyperspace $C(G)$ of all connected compact sets, by dynamical system $(G,f)$ acting on a topological graph $G$. In the case of the base space being a…

We show that the typical nonexpansive mapping on a small enough subset of a CAT($\kappa$)-space is a contraction in the sense of Rakotch. By typical we mean that the set of nonexpansive mapppings without this property is a $\sigma$-porous…

We investigate the topological entropy of operators. More precisely, in the Banach space setting, we show that compact operators have finite entropy, which depends solely on their point spectrum. Moreover, for operators on \(F\)-spaces, we…

We construct a family $\{\Phi_t\}_{t\in[0,1]}$ of homeomorphisms of the two-torus isotopic to the identity, for which all of the rotation sets $\rho(\Phi_t)$ can be described explicitly. We analyze the bifurcations and typical behavior of…

Given an extended real-valued submeasure $\nu$ defined on a field of subsets $\Sigma$ of a given set, we provide necessary and sufficient conditions for which the pseudometric $d_\nu$ defined by $d_{\nu}(A,B):=\min\{1,\nu(A\bigtriangleup…

In this paper, we establish a new quasi-shadowing property for any nonuiformly partially hyperbolic set of a $C^{1+\alpha}$ diffeomorphism, which is adaptive to the movement of the pseudo-orbit. Moreover, the quasi-specification property…

We establish an epsilon-regularity theorem at points in the free boundary of almost-minimizers of the energy $\mathrm{Per}_{w}(E)=\int_{\partial^*E}w\,\mathrm{d} {\mathscr{H}}^{n-1}$, where $w$ is a weight asymptotic to…

We show that every intermediate $\beta$-transformation is topologically conjugate to a greedy $\beta$-transformation with a hole at zero, and provide a counterexample illustrating that the correspondence is not one-to-one. This…

Let $(X,d)$ be a compact metric space, $f:X\rightarrow X$ be a continuous transformation with the specification property. we consider non-dense orbit set $E(z_0)$ and show that for any non-transitive point $z_0\in X$, this set $E(z_0)$ is…

Let $f$ be a partially hyperbolic diffeomorphism on a closed (i.e., compact and boundaryless) Riemannian manifold $M$ with a uniformly compact center foliation $\mathcal{W}^{c}$. The relationship among topological entropy $h(f)$, entropy of…

We show that if the maximum modulus of a quasiregular mapping f grows sufficiently rapidly then there exists a non-empty escaping set I(f) consisting of points whose forward orbits under iteration tend to infinity. This set I(f) has an…

This paper introduces the concept of quasi $\alpha$-firmly nonexpansive mappings in Wasserstein spaces over $\mathbb R^d$ and analyzes properties of these mappings. We prove that for quasi $\alpha$-firmly nonexpansive mappings satisfying a…

In the setting of a doubling metric measure space, we study regularity of sets with finite $s$-perimeter, that is, sets whose characteristic functions have finite Besov energy, with regularity parameter $0<s<1$ and exponent $p=1$. Following…

Given a bounded domain $\Omega \subset {\Bbb R}^d$ with positive measure and a finite set $A=\{a^1, a^2, \dots, a^d\}$, we say that the set ${\mathcal E}(A)={\{e^{2 \pi i x \cdot a^j}\}}_{a^j \in A}$ is a complete exponential system if for…

Conditions are given which imply that certain non-autonomous analytic iterated function systems (NIFS's) in the complex plane have uniformly perfect attractor sets, while other conditions imply the attractor is pointwise thin, and thus…

We consider a continuous dynamical system $f:X\to X$ on a compact metric space $X$ equipped with an $m$-dimensional continuous potential $\Phi=(\phi_1,\cdots,\phi_m):X\to \bR^m$. We study the set of ground states $ GS(\alpha)$ of the…

A famous theorem of Carleson says that, given any function $f\in L^p(\TT)$, $p\in(1,+\infty)$, its Fourier series $(S_nf(x))$ converges for almost every $x\in \mathbb T$. Beside this property, the series may diverge at some point, without…