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Register a new userDegree Complexity of a Family of Birational Maps
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We compute the degree complexity of a family of birational mappings of the plane with high order singularities.
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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.
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 construct a family of birational maps acting on two dimensional projective varieties, for which the growth of the degrees of the iterates is cubic. It is known that this growth can be bounded, linear, quadratic or exponential for such maps acting on two dimensional compact Kahler varieties. The example we construct goes beyond this limitation, thanks to the presence of a singularity on the variety where the maps act. We provide all details of the calculations.
We examine iteration of certain skew-products on the bidisk whose components are rational inner functions, with emphasis on simple maps of the form $Phi(z_1,z_2) = (phi(z_1,z_2), z_2)$. If $phi$ has degree $1$ in the first variable, the dynamics on each horizontal fiber can be described in terms of Mobius transformations but the global dynamics on the $2$-torus exhibit some complexity, encoded in terms of certain $mathbb{T}^2$-symmetric polynomials. We describe the dynamical behavior of such mappings $Phi$ and give criteria for different configurations of fixed point curves and rotation belts in terms of zeros of a related one-variable polynomial.
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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