动力系统
A hyperbolic transcendental entire function with connected Fatou set is said to be of disjoint type. It is known that the Julia set of a disjoint-type function of finite order is a Cantor bouquet; in particular, it is a collection of arcs…
We develop a general technique for realising full closed subsets of the complex plane as wandering sets of entire functions. Using this construction, we solve a number of open problems. (1) We construct a counterexample to Eremenko's…
Simple conditions have been developed in [Zhang, Wahl and Yu, SIAM Rev. 2014; Yu and Wang, Math. Biosci. Eng. 2019], which are used to identify the existence of slow-fast relaxation oscillations that appear in differential systems, where…
This paper investigates recurrence properties of dynamical systems under the restriction that control is available only through inputs and outputs. We introduce the concept of ``coarse non-wandering'', a generalization of the classical…
Consider the geodesic flow on a closed rank one manifold of nonpositive curvature. For certain H\"{o}lder continuous potential, there exists a unique equilibrium state by \cite{BCFT}. In this paper, we introduce the notions of core limit…
We introduce a class of codes with overlapping code words, that we call SPO-codes. The SPO-codes are related to the Markov codes that were introduced in: G. Keller, J. Combinatorial Theory 56, (1991),pp.\ 75--83. The process of generating a…
We determine weak asymptotics of counting functions on generic surfaces in a component of a stratum of $k$-differentials when $k$ is prime and genus is greater than $2$. In order to do so, we classify the $GL^+(2,\mathbb{R})$-orbit closure…
We describe a primary limb structure in the connectedness locus of complex cubic polynomials, where the limbs are indexed by the periodic points of the doubling map $t \mapsto 2t \ (\operatorname{mod} {\mathbb Z})$. The main renormalization…
Let $G$ be a connected semisimple real algebraic group. The class of transverse subgroups of $G$ includes all discrete subgroups of rank one Lie groups and any subgroups of Anosov or relative Anosov subgroups. Given a transverse subgroup…
In this paper we study homeomorphisms of the circle with several critical points and bounded type rotation number. We prove complex a priori bounds for these maps. As an application, we get that bi-cubic circle maps with same bounded type…
Natural disasters, such as hurricanes and typhoons, pose significant challenges to public safety and infrastructure. While government agencies rely on multi million dollar UAV systems for storm data collection and disaster response, smaller…
The aim of the present paper is to propose and study a dissipative variant of symplectic billiards within planar strictly convex domains. The associated billiard map is dissipative, thus it admits a compact invariant set, the so-called…
Numerous studies have reported two types of doubling of invariant closed curves (ICCs) in dynamical systems: (a) the creation of two disjoint ICCs such that iterations flip between them; and (b) the creation of a single ICC of double the…
Chaotic phases in stochastic differential equations are characterized by two essential long-time dynamical features: a random attractor capturing asymptotic geometry and a Sinai-Ruelle-Bowen (SRB) measure describing statistical information.…
We study natural one-parameter families of antiholomorphic correspondences arising from univalent restrictions of Shabat polynomials, indexed by rooted dessin d'enfants. We prove that the parameter spaces are topological quadrilaterals,…
We constructed a DA on $\mathbb{T}^3$, which complements the work of Gan, Li, Viana, and Yang (\cite{GanLiVianaYang2021}) by providing an example of a $C^\infty$-diffeomorphism with partial volume expansion, where $\dim(E^{cs}) = 2$. In…
We construct a continuous linear cocycle over an expanding base dynamics for which the Lyapunov exponents of all ergodic invariant probability measures are small, except for one measure whose Lyapunov exponents are away from zero. The…
The dynamics and statistical properties of two-dimensional (2D) turbulence are often investigated through numerical simulations of incompressible, viscous fluids in doubly periodic domains. A key challenge in 2D turbulence research is…
We study algebraic relations among postcritically finite (PCF) parameters in the family $f_c(z) = z^2 + c$. Ghioca, Krieger, Nguyen and Ye proved that an algebraic curve in $\mathbb{C}^2$ contains infinitely many PCF pairs $(c_1, c_2)$ if…
We consider the set of points with high pointwise emergence for $C^{1+\alpha}$ diffeomorphisms preserving a hyperbolic measure. We find a lower bound on the Hausdorff dimension of this set in terms of unstable Hausdorff dimension of the…