Tien-Cuong Dinh, Nessim Sibony
Let f be a meromorphic correspondence on a compact Kahler manifold. We show that the topological entropy of f is bounded from above by the logarithm of its maximal dynamical degree. An analogous estimate for the entropy on subvarieties is given. We also discuss a notion of Julia and Fatou sets.
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Let $f$ be a meromorphic correspondence on a compact Kähler manifold $X$ of dimension $k$. Assume that its topological degree is larger than the dynamical degree of order $k-1$. We obtain a quantitative regularity of the equilibrium measure of $f$ in terms of its super-potentials.
This paper studies recurrence phenomena in iterative holomorphic dynamics of certain multi-valued maps. In particular, we prove an analogue of the Poincaré recurrence theorem for meromorphic correspondences with respect to certain dynamically interesting measures associated with them. Meromorphic correspondences present a significant measure-theoretic obstacle: the image of a Borel set under a meromorphic correspondence need not be Borel. We manage this issue using the Measurable Projection Theorem, which is an aspect of descriptive set theory. We also prove a result concerning invariance prop…
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, we study the particular case of a family of generic birational maps of P^k for which we construct the Green currents and the equilibrium measure. We use for that the theory of super-potentials. We show that the measure is mixing and gives no mass to pluripolar sets. Using the criterion we get that the measure is of maximal entropy. It implies finally that the…
Let $f : X\to X$ be a dominating meromorphic map on a compact Kähler manifold $X$ of dimension $k$. We extend the notion of topological entropy $h^l_{\mathrm{top}}(f)$ for the action of $f$ on (local) analytic sets of dimension $0\leq l \leq k$. For an ergodic probability measure $ν$, we extend similarly the notion of measure-theoretic entropy $h_ν^l(f)$. Under mild hypothesis, we compute $h^l_{\mathrm{top}}(f)$ in term of the dynamical degrees of $f$. In the particular case of endomorphisms of $\mathbb{P}^2$ of degree $d$, we show that $h^1_{\mathrm{top}}(f)= \log d$ for a large class of maps…
A meromorphic transform between complex manifolds is a surjective mutivalued map with an analytic graph. Let $F_n$ be a sequence of meromorphic transforms from a compact Kahler manifold X into compact Kahler manifolds X_n. We give conditions which imply that the behavior of the sequence of preimages $F_n^{-1}(x_n)$ of x_n does not depend on the generic sequence of points (x_1,x_2,....). Using this formalism, we obtain sharp results on the limit distribution of common zeros, of $l$ random holomorphic sections of high powers $L^n$ of a positive holomorphic line bundle L over a projective manifol…
Let $X$ and $Y$ be compact Kähler manifolds, and let $f:X\rightarrow Y$ be a dominant meromorphic map. Base upon a regularization theorem of Dinh and Sibony for DSH currents, we define a pullback operator $f^{\sharp}$ for currents of bidegrees $(p,p)$ of finite order on $Y$ (and thus for {\it any} current, since $Y$ is compact). This operator has good properties as may be expected. Our definition and results are compatible to those of various previous works of Meo, Russakovskii and Shiffman, Alessandrini and Bassanelli, Dinh and Sibony, and can be readily extended to the case of meromorphic co…
We construct a space which is useful in order to study the entropy of meromorphic maps by using projective limits. We deduce a variational principle for meromorphic maps.
Correspondences on hyperelliptic surfaces, combination theorems, and Hurwitz spaces
本文构造了任意亏格双曲黎曼曲面上结合零亏格富克斯群与布拉施克乘积的对应，给出其解析组合的代数刻画，识别对应模空间为泰希米勒空间与布拉施克空间的乘积，并构造该乘积空间到赫维茨空间的动态自然单射。
We show that any dominant meromorphic self-map f of a compact Kaehler manifold X is an Artin-Mazur map. More precisely, if P_n(f) is the number of its isolated periodic points of period n (counted with multiplicity), then P_n(f) grows at most exponentially fast with respect to n and the exponential rate is at most equal to the algebraic entropy of f. Further estimates are given when X is a surface. Among the techniques introduced in this paper, the h-dimension of the density between two arbitrary positive closed currents on a compact Kaehler surface is obtained.
We consider the dynamics of a meromorphic map on a compact kahler surface whose topological degree is smaller than its first dynamical degree. The latter quantity is the exponential rate at which its iterates expand the cohomology class of a kahler form. Our goal in this article and its sequels is to carry out a conjectural program for constructing and analyzing a natural measure of maximal entropy for each such map. Here we take the first step, converting information about the linear action of the map on cohomology to invariant currents with special geometric structure. We also give some exam…
We continue our study of the dynamics of mappings with small topological degree on (projective) complex surfaces. Previously, under mild hypotheses, we have constructed an ergodic ``equilibrium'' measure for each such mapping. Here we study the dynamical properties of this measure in detail: we give optimal bounds for its Lyapunov exponents, prove that it has maximal entropy, and show that it has product structure in the natural extension. Under a natural further assumption, we show that saddle points are equidistributed towards this measure. This generalize results that were known in the inve…
We study two notions of topological entropy of correspondences introduced by Friedland and Dinh-Sibony. Upper bounds are known for both. We identify a class of holomorphic correspondences whose entropy in the sense of Dinh-Sibony equals the known upper bound. This provides an exact computation of the entropy for rational semigroups. We also explore a connection between these two notions of entropy.
We prove a theorem of uppersemicontinuity for the metric entropy of meromorphic maps.
Let $f:X\to X$ be a dominating meromorphic map of a compact Kähler surface of large topological degree. Let $S$ be a positive closed current on $X$ of bidegree $(1,1)$. We consider an ergodic measure $ν$ of large entropy supported by $\mathrm{supp}(S)$. Defining dimensions for $ν$ and $S$, we give inequalities à la Mañé involving the Lyapunov exponents of $ν$ and those dimensions. We give dynamical applications of those inequalities.
We extend the Dinh-Sibony notion of densities of currents to the setting where the ambient manifold is not necessarily Kähler and study the intersection of analytic sets from the point of view of densities of currents. As an application, we introduce the notion of exotic periodic points of a meromorphic self-map. We then establish the expected asymptotic for the sum of the number of isolated periodic points and the number of exotic periodic points for holomorphic self-maps with a simple action on the cohomology groups on a compact Kähler manifold. We also show that the algebraic entropy of mer…
We use Bowen's definition of topological entropy and Ahlfors five islands theorem, as well as the theory of polynomial-like mappings, to show that the topological entropy of any entire transcendental function is infinity. In addition the entropy is concentrated on the Julia set for each meromorphic function which has no wandering domains.
Periodic points for meromorphic self-maps of Fujiki varieties
本文研究Fujiki类紧复流形上支配亚纯自映射的周期点增长，证明当拓扑次数严格大于其他动力学次数时孤立周期点数量呈指数增长，从而肯定Shou-Wu Zhang的猜想，并在一般情形下给出其指数增长的上界为代数熵。
Let T be a positive closed (p,p)-current of mass 1 on a compact Kahler manifold X. Then, there exist a constant c, independent of T, and smooth positive closed (p,p)-currents Tn and Sn of mass c such that Tn-Sn converge weakly to T. We also extend this result to positive pluriharmonic currents. Then we study the wedge product of positive closed (1,1)-currents having continuous potential with positive pluriharmonic currents. As an application, we give an estimate of the topological entropy of meromorphic maps on compact Kahler manifolds.
Let f be a dominating meromorphic self-map of a compact Kahler manifold. We give an inequality for the Lyapounov exponents of some ergodic measures of f using the metric entropy and the dynamical degrees of f. We deduce the hyperbolicity of some measures.
Let f be a dominant rational map of P^k such that there exists s <k, with lambda_s(f)>lambda_l(f) for all l. Under mild hypotheses, we show that, for A outside a pluripolar set of the group of automorphisms of P^k, the map f o A admits a hyperbolic measure of maximal entropy log(lambda_s(f)) with explicit bounds on the Lyapunov exponents. In particular, the result is true for polynomial maps hence for the homogeneous extension of f to P^{k+1}. This provides many examples where non uniform hyperbolic dynamics is established. One of the key tools is to approximate the graph of a meromorphic func…
Let $X$ be a compact complex manifold of dimension $k$ and $f:X \longrightarrow X$ be a dominating meromorphic map. We generalize the notion of topological entropy, by defining a quantity $h_{(m,l)}^{top}(f)$ which measures the action of $f$ on local analytic sets $W$ of dimension $l$ with $W \subset f^{-n}(Δ)$ where $Δ$ is a local analytic set of dimension $m$. We give then inequalities between $h_{(m,l)}^{top}(f)$ and Lyapounov exponents of suitable invariant measures.
MME for Mating Correspondences
利用Vivas-Sirvent的测度熵和Dinh-Sibony、Kelly-Tennant的拓扑熵，证明作者发现的均衡测度μ_-和μ_+最大化Bullett-Lomonaco研究的单参数族{F_a}_a的熵，并找到Bullett研究的两个删除覆盖对应复合的最大熵测度。
A variational principle for holomorphic correspondences
本文提出全纯对应关系的变分原理，通过定义测度论熵和压力来研究动力系统的行为。
Motivated by recent applications to entropy theory in dynamical systems, we generalise notions introduced by Matthews and define weakly weighted and componentwisely weakly weighted (generalised) quasi-metrics. We then systematise and extend to full generality the correspondences between these objects and other structures arising in theoretical computer science and dynamics. In particular, we study the correspondences with weak partial metrics, and, if the underlying space is a semilattice, with invariant (generalised) quasi-metrics satisfying the descending path condition, and with strictly mo…
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