Hyperbolic measure of maximal entropy for generic rational maps of ℙ k

Gabriel Vigny[1]

  • [1] U. P. J. V. LAMFA - UMR 7352 33, rue Saint-Leu 80039 Amiens (France)

Annales de l’institut Fourier (2014)

  • Volume: 64, Issue: 2, page 645-680
  • ISSN: 0373-0956

Abstract

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Let f be a dominant rational map of ℙ k such that there exists s < k with λ s ( f ) > λ l ( f ) for all l . Under mild hypotheses, we show that, for A outside a pluripolar set of Aut ( ℙ k ) , the map f ∘ A admits a hyperbolic measure of maximal entropy log λ 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 function by a smooth positive closed current. This allows us to do all the computations in a smooth setting, using super-potentials theory to pass to the limit.

How to cite

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Vigny, Gabriel. "Hyperbolic measure of maximal entropy for generic rational maps of $\mathbb{P}^k$." Annales de l’institut Fourier 64.2 (2014): 645-680. <http://eudml.org/doc/275576>.

@article{Vigny2014,
abstract = {Let $f$ be a dominant rational map of $\mathbb\{P\}^k$ such that there exists $s &lt;k$ with $\lambda _s(f)&gt;\lambda _l(f)$ for all $l$. Under mild hypotheses, we show that, for $A$ outside a pluripolar set of $\mathrm\{Aut\} (\mathbb\{P\}^k)$, the map $f\circ A$ 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 $\mathbb\{P\}^\{k+1\}$. This provides many examples where non uniform hyperbolic dynamics is established.One of the key tools is to approximate the graph of a meromorphic function by a smooth positive closed current. This allows us to do all the computations in a smooth setting, using super-potentials theory to pass to the limit.},
affiliation = {U. P. J. V. LAMFA - UMR 7352 33, rue Saint-Leu 80039 Amiens (France)},
author = {Vigny, Gabriel},
journal = {Annales de l’institut Fourier},
keywords = {Complex dynamics; meromorphic maps; Super-potentials; entropy; hyperbolic measure; holomorphic dynamics; super-potentials},
language = {eng},
number = {2},
pages = {645-680},
publisher = {Association des Annales de l’institut Fourier},
title = {Hyperbolic measure of maximal entropy for generic rational maps of $\mathbb\{P\}^k$},
url = {http://eudml.org/doc/275576},
volume = {64},
year = {2014},
}

TY - JOUR
AU - Vigny, Gabriel
TI - Hyperbolic measure of maximal entropy for generic rational maps of $\mathbb{P}^k$
JO - Annales de l’institut Fourier
PY - 2014
PB - Association des Annales de l’institut Fourier
VL - 64
IS - 2
SP - 645
EP - 680
AB - Let $f$ be a dominant rational map of $\mathbb{P}^k$ such that there exists $s &lt;k$ with $\lambda _s(f)&gt;\lambda _l(f)$ for all $l$. Under mild hypotheses, we show that, for $A$ outside a pluripolar set of $\mathrm{Aut} (\mathbb{P}^k)$, the map $f\circ A$ 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 $\mathbb{P}^{k+1}$. This provides many examples where non uniform hyperbolic dynamics is established.One of the key tools is to approximate the graph of a meromorphic function by a smooth positive closed current. This allows us to do all the computations in a smooth setting, using super-potentials theory to pass to the limit.
LA - eng
KW - Complex dynamics; meromorphic maps; Super-potentials; entropy; hyperbolic measure; holomorphic dynamics; super-potentials
UR - http://eudml.org/doc/275576
ER -

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