Dynamique des applications d'allure polynomiale

By T. Dinh, N. Sibony. 2002

147 citations recorded by the source corpus.

We study the dynamics of polynomial-like mappings in several variables. A special case of our results is the following theorem: Let f:U→V be a proper holomorphic map from an open set U⋐V onto a Stein manifold V. Assume f is of topological degree dt⩾2. Then there is a probability measure μ supported on K:=⋂n⩾0f−n(V) satisfying the following properties: (1) The measure μ is invariant, K-mixing, of maximal entropy logdt. (2) If J is the Jacobian of f with respect to a volume form Ω then ∫logJdμ⩾logdt. (3) For every probability measure ν on V with no mass on pluripolar sets dt−n(fn)∗ν⇀μ. (4) If the p.s.h. functions on V are μ-integrables (μ is PLB), then (a) The Lyapounov exponents for μ are strictly positive; (b) μ is exponentially mixing; (c) There is a proper analytic subset E0 of V such that f−1(E0)⊂E0 and for z∉E, μzn:=dt−n(fn)∗δz⇀μ where E=⋃n⩾0fn(E0); (d) The measure μ is a limit of Dirac masses on the repelling periodic points. The condition μ is PLB is stable under small pertubation of f. This gives large families where it is satisfied.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/64a99ca2a9e40f164afa41ef6ccbec5884ccfc5c

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Citations in the corpus

Normalization of bundle holomorphic contractions and applications to dynamics (2007, background): “The dynami alproperties of these maps have been studied by Dinh-Sibony [DS℄.”
Une borne supérieure pour l'entropie topologique d'une application rationnelle (2003, background): “Rappelons quelques notions (voir par exemple [1], [18], [17], [7], [12]).”
Green currents for holomorphic automorphisms of compact Kähler manifolds (2004, methodology): “In [13], we showed that a positive measure μ on a Riemann surface admits locally a bounded potential if and only if locally p.”
Dynamics of regular birational maps in P^k (2004, methodology): “The spaces of currents we use as in [7, 9, 10] are probably of interest: they allow to consider intersections of currents of bidegree (p, p), p > 1 (see Remark 2.”
Bifurcation Currents in Holomorphic Families of Rational Maps (2012, background): “In [22], Dinh and Sibony established that L.”
HÉNON-LIKE MAPPINGS IN C² (2004, methodology): “We have to point out at this stage that T. Dinh and N. Sibony [ DiS ] recently studied polynomial-like maps in several variables, obtaining results which parallel properties of holomorphic self maps of P k .,We use some convexity properties (“subharmonic variation”) of positive currents along with a”
Equidistribution speed for endomorphisms of projective spaces (2009, methodology): “set Eand some extra properties which are useful in the proof. This exceptional set is still not well understood. Very recently, Briend-Duval [2] showed that using a dynamical argument, as in our work [5, 7, 9], one can obtain a short proof of Corollary 1.4. In this paper, we will use known propertie”
Suites d’Applications Méromorphes Multivaluées et Courants Laminaires (2005, background): “Donc elle converge vers un courant S∞. On obtient alors des r´esultats de Lyubich [21], Freire-Lopes-Ma˜n´e [16], Briend-Duval [4] et [ 9 , 19, 8, 10].”
Dynamics of meromorphic maps with small topological degree III: geometric currents and ergodic theory (2008, result): “Let us recall that the ergodic theory of mappings with large topological degree (l2 > l1) has been extensively studied, and that results analogous to our Theorems B and C are true in this context [BrDu, DS , G1].”
Super-potentials of positive closed currents, intersection theory and dynamics (2007, background): “It is easy to check that for a single measure ν, cap(ν) > 0 if and only if quasi-psh functions are ν-integrable, i.e. ν is PB in the sense of [ 20 , 24].”
Bifurcation currents in holomorphic dynamics on ${\bf P}^k$ (2005, background): “on Hd(C), as it has been proved in the larger setting of polynomial like mappings (see [10]).”
Equidistribution problems of complex dynamics in higher dimension (2016, methodology): “n the case where X is a projective manifold, the strategy described in the introduction works quite well. The construction used in the original proof of this theorem follows the strategy developed in [25], which is based on the construction of the inverse branches of fn deﬁned on balls in X. The con”
Exponential estimates for plurisubharmonic functions and stochastic dynamics (2008, background): “, and for ψ Hölder continuous, see [17, 8, 10, 11].”
Pull-back of currents by holomorphic maps (2006, unknown): “Dsh currents have been introduced in [9, 12, 13].”
Bifurcation currents in holomorphic dynamics on (2007, background): “on Hd(Ck+1), as it has been proved in the larger setting of polynomial like mappings (see [ 10 ]).”