Related papers: Slowly recurrent Collet-Eckmann maps on the Rieman…

We study the exponential maps induced by Sobolev type right-invariant (weak) Riemannian metrics of order $k\ge1$ on the Lie group of smooth, orientation preserving diffeomorphisms of the circle. We prove that each of them defines an {\em…

We investigate the dynamics of semigroups of rational maps on the Riemann sphere. To establish a fractal theory of the Julia sets of infinitely generated semigroups of rational maps, we introduce a new class of semigroups which we call…

We investigate the random dynamics of polynomial maps on the Riemann sphere and the dynamics of semigroups of polynomial maps on the Riemann sphere. In particular, the dynamics of a semigroup $G$ of polynomials whose planar postcritical set…

We investigate the iterative behaviour of continuous order preserving subhomogeneous maps that map a polyhedral cone into itself. For these maps we show that every bounded orbit converges to a periodic orbit and, moreover, that there exists…

We associate to each non-degenerate smooth interval map a number measuring its global asymptotic expansion. We show that this number can be calculated in various different ways. A consequence is that several natural notions of nonuniform…

We investigate the question of whether or not the orbit of a point in A/Q, under the natural action of a subset S of Q, is dense in A/Q. We prove that if the set S is a multiplicative semigroup which contains at least two multiplicatively…

We consider rational maps $f$ on the Riemann sphere $\widehat {\mathbb{C}}$ with an $f$-invariant set $P\subset \widehat {\mathbb{C}}$ of four marked points containing the postcritical set of $f$. We show that the dynamics of the…

Recently continuous rational maps between real algebraic varieties have attracted the attention of several researchers. In this paper we continue the investigation of approximation properties of continuous rational maps with values in…

We discuss the asymptotic behaviour of random critical Boltzmann planar maps in which the degree of a typical face belongs to the domain of attraction of a stable law with index $\alpha \in (1,2]$. We prove that when conditioning such maps…

We present a number of rigidity results concerning holomorphic dynamical systems admitting rotation quasicircles. Firstly, we show the absence of line fields on the Julia set of any rational map that is geometrically finite away from a…

In this article, we consider a counting problem for orbits of hyperbolic rational maps on the Riemann sphere, where constraints are placed on the multipliers of orbits. Using arguments from work of Dolgopyat, we consider varying and…

In this article, we study the dynamics of the following family of rational maps with one parameter: \begin{equation*} f_\lambda(z)= z^n+\frac{\lambda^2}{z^n-\lambda}, \end{equation*} where $n\geq 3$ and $\lambda\in\mathbb{C}^*$. This family…

We define the group of almost periodic diffeomorphisms on $\mathbb{R}^n$ and on an arbitrary Lie group. We then study the properties of its Riemannian and Lie group exponential maps and provide applications to fluid equations. In…

We prove a version of the classical $\lambda$-lemma for holomorphic families of Riemann surfaces. We then use it to show that critical loci for complex H\'{e}non maps that are small perturbations of quadratic polynomials with Cantor Julia…

Given a rank-two sub-Riemannian structure $(M,\Delta)$ and a point $x_0\in M$, a singular curve is a critical point of the endpoint map $F:\gamma\mapsto\gamma(1)$ defined on the space of horizontal curves starting at $x_0$. The typical…

A set $A \subseteq \mathbb{N}$ is a set of pointwise recurrence if for all minimal dynamical systems $(X, T)$, all $x \in X$, and all open neighborhoods $U \subseteq X$ of $x$, there exists a time $n \in A$ such that $T^n x \in U$. The set…

A Lorenz map is a Poincar\'e map for a three-dimensional Lorenz flow. We describe the theory of renormalization for Lorenz maps with a critical point and prove that a restriction of the renormalization operator acting on such maps has a…

We give formulas for the density of the measure of orthogonality for orthonormal polynomials with unbounded recurrence coefficients. The formulas involve limits of appropriately scaled Tur\'an determinants or Christoffel functions. Exact…

We investigate the dynamics of semigroups generated by a family of polynomial maps on the Riemann sphere such that the postcritical set in the complex plane is bounded. The Julia set of such a semigroup may not be connected in general. We…

Let $\mathcal{C}(S^{m})$ denote the set of continuous maps from the unit sphere $S^{m}$ in $\mathbb{R}^{m+1}$ into itself endowed with the supremum norm. We prove that the set $\{f^n: f\in \mathcal{C}(S^{m})~\text{and}~n\ge 2\}$ of iterated…