Related papers: Polynomial diffeomorphisms of C^2, IV: The measure…

Let f be a non-invertible holomorphic endomorphism of P^k having an attracting set A. We show that, under some natural assumptions, A supports a unique invariant positive closed current \tau, of the right bidegree and of mass 1. Moreover,…

In this paper we develop the theory of {\it polymorphisms} of measure spaces, which is a generalization of the theory of measure-preserving transformations; we describe the main notions and discuss relations to the theory of Markov…

For a $C^{r}$ $(r>1)$ diffeomorphism on a compact manifold that admits a dominated splitting, this paper establishes the upper semi-continuity of the entropy map. More precisely, this paper establishes the upper semi-continuity of the…

Let $X$ be a compact K\"ahler manifold of dimension 3 and let $f:X\rightarrow X$ be a pseudo-automorphism. Under the mild condition that $\lambda_1(f)^2>\lambda_2(f)$, we prove the existence of invariant positive closed $(1,1)$ and $(2,2)$…

We show that two automorphisms of an affine surface with dynamical degree strictly larger than 1 share a Zariski dense set of periodic points if and only if they have the same periodic points. We construct canonical heights for these…

In this paper, we construct various examples of holomorphic laminations, with leaves of dimension 1, and we also study some of their dynamical properties. In particular we study existence and uniqueness of positive closed currents. We…

We study the structure and the Lyapunov exponents of the equilibrium measure of endomorphisms of $\mathbb P^k$ preserving a fibration. We extend the decomposition of the equilibrium measure obtained by Jonsson for polynomial skew products…

We consider partially hyperbolic diffeomorphisms on compact manifolds where the unstable and stable foliations stably carry some unique non-trivial homologies. We prove the following two results: if the center foliation is one dimensional,…

In this paper, we show that for several interesting systems beyond uniform hyperbolicity, any generic continuous function has a unique maximizing measure with zero entropy. In some cases, we also know that the maximizing measure has full…

In this note, we study monotone dynamical systems with respect to polyhedral cones. Using the half-space representation and the vertex representation, we propose three equivalent conditions to certify monotonicity of a dynamical system with…

In this paper we study the ergodic theory of a class of symbolic dynamical systems $(\O, T, \mu)$ where $T:{\O}\to \O$ the left shift transformation on $\O=\prod_0^\infty\{0,1\}$ and $\mu$ is a $\s$-finite $T$-invariant measure having the…

We study the partially hyperbolic diffeomorphims whose center direction admits the u-definite property in the sense that all the central Lyapunov exponents of each ergodic Gibbs u-state are either all positive or all negative. We prove that…

We develop a general framework for establishing non-uniqueness of stationary measures for stochastically forced dynamical systems possessing an almost surely invariant submanifold. Our main abstract result provides sufficient conditions for…

We show that the Lattes endomorphisms are the only holomorphic endomorphisms of the complex k-dimensional projective space whose measure of maximal entropy is absolutely continuous with respect to the Lebesgue measure. As a consequence,…

We develop a new method, based on pluripotential theory, to study the transfer (Perron-Frobenius) operator induced on $\mathbb P^k = \mathbb P^k (\mathbb C)$ by a holomorphic endomorphism and a suitable continuous weight. This method allows…

On a closed hyperbolic surface, we investigate semiclassical defect measures associated with the magnetic Laplacian in the presence of a constant magnetic field. Depending on the energy level where the eigenfunctions concentrate, three…

Given a surface $M$ and a Borel probability measure $\nu$ on the group of $C^2$-diffeomorphisms of $M$, we study $\nu$-stationary probability measures on $M$. Assuming the positivity of a certain entropy, the following dichotomy is proved:…

We study how physical measures vary with the underlying dynamics in the open class of $C^r$, $r>1$, strong partially hyperbolic diffeomorphisms for which the central Lyapunov exponents of every Gibbs $u$-state is positive. If transitive,…

We continue our study of the dynamics of meromorphic mappings with small topological degree on a compact K\"ahler surface $X$. Under general hypotheses we are able to construct a canonical invariant measure which is mixing, does not charge…

In this paper we study the sequence of orthonormal polynomials $\{P_n(\mu; z)\}$ defined by a probability measure $\mu$ with non-polar compact support $S(\mu)\subset\mathbb C$. We show that the support of any weak* limit of the sequence of…