Related papers: A Characterization of hyperbolic potentials of rat…

Let $\Lambda$ be a quasi-projective variety and assume that, either $\Lambda$ is a subvariety of the moduli space $\mathcal{M}_d$ of degree $d$ rational maps, or $\Lambda$ parametrizes an algebraic family $(f_\lambda)_{\lambda\in\Lambda}$…

In this article, we consider hyperbolic rational maps restricted on thier Julia sets and study about the recurrence rate of typical orbits in arbitrarily small neighbourhoods around them and their relationship to the Hausdorff dimension of…

Let X be a complex projective manifold and f a dominating rational map from X onto X. We show that the topological entropy h(f) of f is bounded from above by the logarithm of its maximal dynamical degree.

We study the dynamics of meromorphic maps for a compact Kaehler manifold X. More precisely, we give a simple criterion that allows us to produce a measure of maximal entropy. We can apply this result to bound the Lyapunov exponents. Then,…

The aim of this paper is to show $J$-stability of immediately expanding rational maps over an algebraically closed, complete, and non-Archimedean field, which is an analogue of R. Man\~e, P. Sad, and D. Sullivan's theorem of $J$-stability…

We show that Fatou components of a semi-hyperbolic rational map are John domains and that the converse does not hold. This generalizes a famous result of Carleson, Jones and Yoccoz. We show that a connected Julia set is locally connected…

A relationship between two old mathematical subjects is observed: the theory of hypergeometric functions and the separability in classical mechanics. Separable potential perturbations of the integrable billiard systems and the Jacobi…

In this article we address the following question, whose interest was recently renewed by problems arising in arithmetic dynamics: under which conditions does there exist a local biholomorphism between the Julia sets of two given…

We prove that any 3-dimensional singular hyperbolic attractor admits for any H\"older continuous potential $V$ at most one equilibrium state for $V$ among regular measures. We give a condition on $V$ which ensures that no singularity can be…

We consider the dynamics arising from the iteration of an arbitrary sequence of polynomials with uniformly bounded degrees and coefficients and show that, as parameters vary within a single hyperbolic component in parameter space, certain…

We characterize two of the most studied non-uniform hyperbolicity conditions for rational maps, semi-hyperbolicity and the topological Collet-Eckmann condition, in terms of the maximal entropy measure. Using the same tools in the proof of…

Let $f_\theta(z)=e^{2\pi i\theta}z+z^2$ be the quadratic polynomial having an indifferent fixed point at the origin. For any bounded type irrational number $\theta\in\mathbb{R}\setminus\mathbb{Q}$ and any rational number $\nu\in\mathbb{Q}$,…

For a class of one-dimensional holomorphic maps f of the Riemann sphere we prove that for a wide class of potentials h the topological pressure is entirely determined by the values of h on the repelling periodic points of f. This is a…

We construct an explicit example of family of non-uniformly hyperbolic diffeomorphisms, at the boundary of the set of uniformly hyperbolic systems, with one orbit of cubic heteroclinic tangency. One of the leaves involved in this…

For a continuous map $f$ on a compact metric space we study the geometry and entropy of the generalized rotation set $\R(\Phi)$. Here $\Phi=(\phi_1,...,\phi_m)$ is a $m$-dimensional continuous potential and $\R(\Phi)$ is the set of all…

We consider partially hyperbolic diffeomorphisms $f$ with a one-dimensional central direction such that the unstable entropy exceeds the stable entropy. Our main result proves that such maps have a finite number of ergodic measures of…

For two positive maps $\phi_i:B(\mathcal{K}_i)\to B(\mathcal{H}_i)$, $i=1,2$, we construct a new linear map $\phi:B(\mathcal{H})\to B(\mathcal{K})$, where $\mathcal{K}=\mathcal{K}_1\oplus\mathcal{K}_2\oplus\mathbb{C}$,…

In this note, we propose a super version of Jacobian conjecture on the automorphisms of affine superspaces over an algebraically closed field $\mathbb{F}$ of characteristic $0$, which predicts that for a homomorphism $\varphi$ of the…

Let $K$ be a finitely generated field of characteristic zero. We study, for fixed $m \geq 2$, the rational functions $\phi$ defined over $K$ that have a $K$-orbit containing infinitely many distinct $m$th powers. For $m \geq 5$ we show the…

We investigate the theory of thermodynamic formalism from the perspective of computable analysis, with a special focus on the computability of equilibrium states. Specifically, we develop two complementary general approaches to verify the…