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 us to prove the existence and uniqueness of the equilibrium state and conformal measure for very general weights (due to Denker-Przytycki-Urbański in dimension 1 and Urbański-Zdunik in higher dimensions, both in the case of H\''older continuous weights). We establish a number of properties of the equilibrium states, including mixing, K-mixing, mixing of all orders, and an equidistribution of repelling periodic points. Our analytic method replaces all distortion estimates on inverse branches with a unique, global, estimate on dynamical currents, and allows us to reduce the dynamical questions to comparisons between currents and their potentials.
We prove that the minimal chain recurrence classes of a holomorphic endomorphism of $\mathbb P^k$ have finitely many connected components. We also obtain results on arbitrary classes. These strong constraints on the topological dynamics in the phase space are all deduced from the associated action on a space of currents.
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 of $\mathbb C^2$. We also show that the sum of the sectional exponents satisfies a Bedford-Jonsson formula when the fibration is linear, and that this function is plurisubharmonic on families of fibered endomorphisms. In particular, the sectional part of the bifurcation current is a closed positive current on the parameter space.
For an algebraic family $(f_t)$ of regular quadratic polynomial endomorphisms of $\mathbb{C}^2$ parametrized by $\mathbb{D}^*$ and degenerating to a Hénon map at $t=0$, we study the continuous (and indeed harmonic) extendibility across $t=0$ of a potential of the bifurcation current on $\mathbb{D}^*$ with the explicit computation of the non-archimedean Lyapunov exponent associated to $(f_t)$. The individual Lyapunov exponents of $f_t$ are also investigated near $t=0$. Using $(f_t)$, we also see that any Hénon map is accumulated by the bifurcation locus in the space of quadratic holomorphic end…
We give an estimate of the discontinuity of the large Julia set for a perturbation of a class of maps tangent to the identity, by means of a two-dimensional Lavaurs Theorem. We adapt to our situation a strategy due to Bedford, Smillie and Ueda in the semiattracting setting. We also prove the discontinuity of the filled Julia set for such perturbations of regular polynomials.
In this paper, we discuss the equidistribution phenomena for holomorphic endomorphisms over $\mathbb{P}^k$ in the case of bidegree $(p,p)$ with $1<p<k$. We prove that if $f:\mathbb{P}^k\to\mathbb{P}^k$ is a holomorphic endomorphism of degree $d\geq 2$ and $T^p$ denotes the Green $(p,p)$-current associated with $f$, then there exists a proper invariant analytic subset $E$ for $f$ such that $d^{-pn}(f^n)^*(S)\to T^p$ exponentially fast in the current sense for every positive closed $(p,p)$-current $S$ of mass 1 such that $S$ is smooth on $E$.
Let $f$ be an endomorphism of $\mathbb{CP}^k$ and $ν$ be an $f$-invariant measure with positive Lyapunov exponents $(λ_1,\...,λ_k)$. We prove a lower bound for the pointwise dimension of $ν$ in terms of the degree of $f$, the exponents of $ν$ and the entropy of $ν$. In particular our result can be applied for the maximal entropy measure $μ$. When $k=2$, it implies that the Hausdorff dimension of $μ$ is estimated by $\dim_{\cal H} μ\geq {\log d \over λ_1} + {\log d \over λ_2}$, which is half of the conjectured formula. Our method for proving these results consists in studying the distribution o…
Let $f$ be an holomorphic endomorphism of $\mathbb{P}^k$ and $μ$ be its measure of maximal entropy. We prove an Almost Sure Invariance Principle for the systems $(\mathbb{P}^k,f,μ)$. Our class $\cal{U}$ of observables includes the Hölder functions and unbounded ones which present analytic singularities. The proof is based on a geometric construction of a Bernoulli coding map $ω: (Σ, s, ν) \to (\mathbb{P}^k,f,μ)$. We obtain the invariance principle for an observable $ψ$ on $(\mathbb{P}^k,f,μ)$ by applying Philipp-Stout's theorem for $χ= ψ\circ ω$ on $(Σ, s, ν)$. The invariance principle implies…
Let $μ$ be the equilibrium measure of an endomorphism of ${\sf P}^k({\bf C})$. We show that it is its unique measure of maximal entropy. We build $μ$ directly as the distribution of any point outside an algebraic exceptional set.
Let $f$ be an endomorphism of $\mathbb{C}\mathbb{P}^k$ of degree $>1$, and assume that for any cyclic Fatou component $W$ of $f$ having a period $p\in\mathbb{N}$, the equilibrium measure $μ_f$ has a positive charge on the boundary of $W$ if and only if $f^{-p}(W)=W$. Then we obtain a locally uniform a priori bound of the dynamics of $f$, which in particular yields a Diophantine-type estimate of the dynamics of $f$ on its domaines singuliers. We also point out that in the case of $k=1$, the statement of our assumption is related to both the impossibility for the Julia set of $f$ to be the bound…
We introduce a notion of stability for equilibrium measures in holomorphic families of endomorphisms of CP(k) and prove that it is equivalent to the stability of repelling cycles and equivalent to the existence of some measurable holomorphic motion of Julia sets which we call equilibrium lamination. We characterize the corresponding bifurcations by the strict subharmonicity of the sum of Lyapunov exponents or the instability of critical dynamics and analyze how repelling cycles may bifurcate. Our methods deeply exploit the properties of Lyapunov exponents and are based on ergodic theory and on…
Let $f$ be an holomorphic endomorphism of $\mathbb{C}\mathbb{P}^k$. We construct by using coding techniques a class of ergodic measures as limits of non-uniform probability measures on preimages of points. We show that they have large metric entropy, close to $\log d^k$. We establish for them strong stochastic properties and prove the positivity of their Lyapunov exponents. Since they have large entropy, those measures are supported in the support of the maximal entropy measure of $f$. They in particular provide lower bounds for the Hausdorff dimension of the Julia set.
The goal of this paper is to study some basic properties of the Fatou and Julia sets for a family of holomorphic endomorphisms of $\mathbb{C}^k,\; k \ge 2$. We are particularly interested in studying these sets for semigroups generated by various classes of holomorphic endomorphisms of $\mathbb{C}^k,\; k \ge 2.$ We prove that if the Julia set of a semigroup $G$ which is generated by endomorphisms of maximal generic rank $k$ in $\mathbb{C}^k$ contains an isolated point, then $G$ must contain an element that is conjugate to an upper triangular automorphism of $\mathbb{C}^k.$ This generalizes a t…
We establish the existence of a spectral gap for the transfer operator induced on $\mathbb P^k = \mathbb P^k (\mathbb C)$ by a generic holomorphic endomorphism and a suitable continuous weight and its perturbations on various functional spaces, which is new even in dimension one. The main issue to overcome is the rigidity of the complex objects, since the transfer operator is a non-holomorphic perturbation of the operator $f_*$. The system is moreover non-uniformly hyperbolic and one may have critical points on the Julia set. The construction of our norm requires the introduction and study of …
Let $f$ be a post-critically finite endomorphism (PCF map for short) on $\mathbb{P}^2$, let $J_1$ denote the Julia set and let $J_2$ denote the support of the measure of maximal entropy. In this paper we show that: 1. $J_1\setminus J_2$ is contained in the union of the (finitely many) basins of critical component cycles and stable manifolds of sporadic super-saddle cycles. 2. For every $x\in J_2$ which is not contained in the stable manifold of a sporadic super-saddle cycle, there is no Fatou disk containing $x$. Here sporadic means that the super-saddle cycle is not contained in a critical co…
LLet $f$ be a holomorphic endomorphism of $\mathbb P^ 2$ of degree $d \geq 2$. We estimate the local directional dimensions of closed positive currents $S$ with respect to ergodic dilating measures $ν$. We infer several applications. The first one shows that the currents $S$ containing a measure of entropy $h\_ν> \log d$ have a directional dimension $>2$, which answers a question by de Thélin-Vigny. The second application asserts that the Dujardin's semi-extremal endomorphisms are close to suspensions of one-dimensional Lattès maps. Finally, we obtain an upper bound for the dimension of the eq…
We initiate a parametric study of holomorphic families of polynomial skew products, i.e., polynomial endomorphisms of $\mathbb{C}^2$ of the form $F(z,w)= (p(z), q(z,w))$ that extend to holomorphic endomorphisms of $\mathbb{P}^2(\mathbb{C})$. We prove that dynamical stability in the sense of arXiv:1403.7603 preserves hyperbolicity within such families, and give a complete classification of the hyperbolic components that are the analogue, in this setting, of the complement of the Mandelbrot set for the family $z^2 +c$. We also precisely describe the geometry of the bifurcation locus and current …
Let f be a holomorphic endomorphism of P^k having an attracting set A. We construct an attracting current and an equilibrium measure associated to A. The attracting current is weakly laminar and extremal in the cone of invariant currents. The equilibrium measure is mixing and has maximal entropy.
A smooth complex variety satisfies the Generalized Jacobian Conjecture if all its étale endomorphisms are proper. We study the conjecture for $\mathbb{Q}$-acyclic surfaces of negative Kodaira dimension. We show that $G$-equivariant counterexamples for infinite group $G$ exist if and only if $G=\mathbb{C}^*$ and we classify them relating them to Belyi-Shabat polynomials. Taking universal covers we get rational simply connected $\mathbb{C}^*$-surfaces of negative Kodaira dimension which admit non-proper $\mathbb{C}^*$-equivariant étale endomorphisms. We prove also that for every integers $r\geq …
