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Register a new userEntropy of meromorphic maps and dynamics of birational maps
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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 measure is hyperbolic.
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We compute the degree complexity of a family of birational mappings of the plane with high order singularities.
We prove the exponential decay of correlations for C^alpha-observables (0<alpha =<2) for generic birational maps of P^k `a la Bedford-Diller. In the particular case of regular birational maps, we give a better estimate of the speed of the decay, getting results as sharp as Dinhs results for Henon maps.
We study bifurcation phenomena in natural families of rational, (transcendental) entire or meromorphic functions of finite type ${f_lambda := varphi_lambda circ f_{lambda_0} circ psi^{-1}_lambda}_{lambdain M}$, where $M$ is a complex connected manifold, $lambda_0in M$, $f_{lambda_0}$ is a meromorphic map and $varphi_lambda$ and $psi_lambda$ are families of quasiconformal homeomorphisms depending holomorphically on $lambda$ and with $psi_lambda(infty)=infty$. There are fundamental differences compared to the rational or entire setting due to the presence of poles and therefore of parameters for which singular values are eventually mapped to infinity (singular parameters). Under mild geometric conditions we show that singular (asymptotic) parameters are the endpoint of a curve of parameters for which an attracting cycle progressively exits de domain, while its multiplier tends to zero. This proves the main conjecture by Fagella and Keen (asymptotic parameters are virtual centers) in a very general setting. Other results in the paper show the connections between cycles exiting the domain, singular parameters, activity of singular orbits and $J$-unstability, converging to a theorem in the spirit of the celebrated result by Ma~{n}e-Sad-Sullivan and Lyubich.
Let $f : Xto X$ be a dominating meromorphic map on a compact Kahler 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 $0leq l leq k$. For an ergodic probability measure $ u$, we extend similarly the notion of measure-theoretic entropy $h_{ u}^l(f)$. 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 but we give examples where $h^1_{mathrm{top}}(f) eq log d$.
We prove that a long iteration of rational maps is expansive near boundaries of bounded type Siegel disks. This leads us to extend Petersens local connectivity result on the Julia sets of quadratic Siegel polynomials to a general case.
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