动力系统

Given an integer $b\ge 3$, let $T_b: [0,1)\to [0,1); x\mapsto bx\pmod 1$ be the expanding map on the unit circle. For any $m\in\mathbb{N}$ and $\omega=\omega^0\omega^1\ldots\in(\left\{0,1,\ldots,b-1\right\}^m)^\mathbb{N_0}$ let \[…

In this paper, we use thermodynamic formalism to study the dynamics of inner functions $F$ acting on the unit disk. If the Denjoy-Wolff point of $F$ is in the open unit disk, then without loss of generality, we can assume that $F(0) = 0$ so…

We study the circle restrictions of inner functions of the unit disc showing that the local invertibility of a restriction is independent of its singularity set and proving a local characterization of analytic conditional expectations. We…

Let $\mu$ be a self-conformal measure on $\mathbb{R}^d$. In this note we establish conditions for $\mu$ under which $\dim(\mu*\nu) = \min\lbrace d,\dim\mu+\dim\nu\rbrace$ holds when $\nu$ is any Ahlfors-regular or self-conformal measure on…

We prove the existence of planar $D_n$--equivariant choreographies in the $n$--body problem with homogeneous potential of degree $-\alpha$, $0<\alpha<2$. Each body follows the same closed path, rotated and time-shifted, forming a…

We consider a class of fast-slow $C^4$ partially hyperbolic systems on $\mathbb{T}^2$ given by $\epsilon$-perturbations of maps $F(x,\theta)=(f(x,\theta),\theta)$ where $f(\cdot,\theta)$ are $C^{4}$ expanding maps of the circle. For…

We study the nonlinear oscillator z'' + omega^2 z + g(t) z^2 = 0 with a time-dependent coefficient g(t). We show that this equation admits an exact quadratic invariant I(z,p,t) provided that g(t) = alpha2(t)^(-5/2) and that alpha2(t)…

The notion of an attractor has various definitions in the theory of dynamical systems. Under compactness assumptions, several of those definitions coincide and the theory is rather complete. However, without compactness, the picture becomes…

In this paper, we study the dynamics of set-valued maps whose graphs are open and such that the image of each point is an open and connected set. Building upon the work of P. Duarte and M. Torres, who introduced and analyzed the…

We show ergodicity of (asymmetric) lemon billiards, billiard tables that are the intersection of two circles of which one contains the centers of both. These do not satisfy the Wojtkowski criteria for hyperbolicity, but we establish…

Let $\Omega$ and $M$ be compact smooth manifolds and let $\Theta:\Omega\times M\to\Omega\times M$ be a $\mathcal C^{1+\alpha}$ skew-product diffeomorphism over a transitive Anosov base. We show that $\Theta$ has at most countably many…

We construct stochastic multisymplectic systems by considering a stochastic extension to the variational formulation of multisymplectic partial differential equations proposed in [Hydon, {\it Proc. R. Soc. A}, 461, 1627--1637, 2005]. The…

In this paper, we build some ergodic theorems involving function $\Omega$, where $\Omega(n)$ denotes the number of prime factors of a natural number $n$ counted with multiplicities. As a combinatorial application, it is shown that for any…

Let $k\in \mathbb Z_+$ and $(X, \mathcal B(X), \mu)$ be a probability space equipped with a family of commuting invertible measure-preserving transformations $T_1,\ldots, T_k \colon X\to X$. Let $P_1,\ldots, P_k\in\mathbb Z[\rm n]$ be…

In this paper, we study mixing rates for $\mathbb{T}^{d}$-extensions of hyperbolic flows. Given three closed orbits with their holonomies, we can relate them to a point in $\mathbb{R}^{d+1}$. We prove that the extension flow enjoys rapid…

In this work we revisit and extend the method introduced by Lins Neto, Sad and Sc\'{a}rdua for detecting the non-existence of invariant algebraic curves other than some prescribed invariant nodal curve. We prove that, under the existence of…

We prove a conjecture of Calta, Kraaikamp and the author: For all $n\ge 3$, each member of their one-parameter family of interval maps, denoted $T_{3,n,\alpha}$, has its `first expansive return map' of natural extension given by the first…

We consider diffeomorphisms $f$ with heterodimensional cycles of co-index two, associated with saddles $P$ and $Q$ having unstable indices $\ell$ and $\ell+2$, respectively. In a partially hyperbolic setting, where a two-dimensional center…

We show that $\mathcal{C}^{\infty}$ local diffeomorphisms of closed surfaces whose topological entropy is larger than the logarithm of their degree admit a finite number of ergodic measures of maximal entropy. To do this, we construct…

We investigate rigidity phenomena associated to the stable norm and Mather's $\beta$-function for Riemannian geodesic flows on closed manifolds. Given two metrics $g_1$ and $g_2$, we compare these objects pointwise at individual homology…