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N. Sibony

Publications and source records attributed to N. Sibony.

6 recordsLinked to original sources
arXiv2008

Dynamics of horizontal-like maps in higher dimension

We study the regularity of the Green currents and of the equilibrium measure associated to a horizontal-like map in C^k, under a natural assumption on the dynamical degrees. We estimate the speed of convergence towards the Green currents, the decay of correlations for the equilibrium measure and the Lyapounov exponents. We show in particular that the equilibrium measure is hyperbolic. We also show that the Green currents are the unique invariant vertical and horizontal positive closed currents. The results apply, in particular, to Henon-like maps, to regular polynomial automorphisms of C^k and to their small pertubations.

math.DS↗
arXiv2003

Value distribution of meromorphic transforms and applications

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 manifold X. We consider also the equidistribution problem for random iteration of correspondences. If f is a meromorphic self correspondence of a compact Kahler manifold X, under a hypothesis on the dynamical degrees, we construct an $f^*$-invariant probability measure $μ$ such that quasi-p.s.h. functions are $μ$-integrable. Every projective manifold admits such correspondences. When f is a meromorphic map, the measure $μ$ is exponentially mixing. We give some analogous results for random iterations of correspondences. We also consider the problem of equidistribution of preimages of subvarieties for a correspondences and more precisely for polynomial automorphisms.

math.DS↗
arXiv2002

Groupes commutatifs d'automorphismes d'une variete Kahlerienne compacte

Let V be a compact Kahler manifold. Let G' be a commutative subgroup of Aut(V) and U the set of elements of zero entropy of G'. Then U is a group and G' is isomorphic to the direct product of groups U and G where G is a subgroup of G' such that all elements of G, except the identity, are of positive entropy. Moreover, G is a free commutative group with rank(G)<dim(V). The estimate is sharp. When rank(G)=dim(V)-1, U is finite. Rank(G) satisfies other inequalities involving the dimensions of Dolbeault cohomology groups of V.

math.DS↗
arXiv2002

Dynamique des applications polynomiales semi-regulieres

For any proper polynomial map $f:C^k\longrightarrow C^k$ define the function αas $$α(z):=\limsup_{n\to\infty} \frac{\log^+\log^+|f^n(z)|}{n} where \log^+:=\max(\log, 0).$$ Let f=(P_1,...,P_k) be a proper polynomial map. We define a notion of s-regularity using the extension of f to P^k. When f is (maximally) regular we show that the function αis l.s.c and takes only finitely many values: 0 and d_1, ..., d_k, where d_i:=deg P_i. We then describe dynamically the sets (α\leq d_i). If d_i>1, this allows us to construct the equilibrium measure μassociated to f as a generalized intersection of positive currents. We then gives an estimate of the Hausdorff dimension of μ. This is a special case of our results. We extend the approach to the larger class of (π,s)-regular maps. This gives an understanding of the biggest values of α. The results can be applied to construct dynamically interesting measures for automorphisms.

math.DS↗
arXiv2002

Dynamique des applications d'allure polynomiale

We study the dynamics of polynomial-like mappings in several variables. A special case of our results is the following theorem. Let f be a proper holomorphic map from an open set U onto a Stein manifold V, $U\subset\subset V$. Assume f is of topological degree d_t>1. Then there is a probability measure μsupported on $\bigcap_{n\geq 0}f^{-n}(V)$ satisfying the following properties. 1. The measure μis invariant, K-mixing, of maximal entropy \log d_t. 2. If J is the Jacobian of f with respect to a volume form then $\int \log J \d μ\geq \log d_t$. 3. For every probability measure νon V with no mass on pluripolar sets $d_t^{-n} (f^n)^*ν$ converges to $μ$. 4. If the p.s.h. functions on V are μ-integrables (μis PLB) then (a) The Lyapounov exponents for μare strictly positive. (b) μis exponentially mixing. (c) There is a proper analytic subset E of V such that for $z\not\in\E$, $μ^z_n:=d_t^{-n} (f^n)^*δ_z$ converges to μ. (d) The measure μis a limit of Dirac masses on the repelling periodic points. The condition μis PLB is stable under small pertubation of f. This gives large families where it is satisfied.

math.DS↗