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Register a new userContinuity of the Green function in meromorphic families of polynomials
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We prove that along any marked point the Green function of a meromorphic family of polynomials parameterized by the punctured unit disk explodes exponentially fast near the origin with a continuous error term.
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We show that for each $din (0,2]$ there exists a meromorphic function $f$ such that the inverse function of $f$ has three singularities and the Julia set of $f$ has Hausdorff dimension $d$.
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.
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.
A function which is transcendental and meromorphic in the plane has at least two singular values. On one hand, if a meromorphic function has exactly two singular values, it is known that the Hausdorff dimension of the escaping set can only be either $2$ or $1/2$. On the other hand, the Hausdorff dimension of escaping sets of Speiser functions can attain every number in $[0,2]$ (cf. cite{ac1}). In this paper, we show that number of singular values which is needed to attain every Hausdorff dimension of escaping sets is not more than $4$.
We initiate a parametric study of holomorphic families of polynomial skew products, i.e., polynomial endomorphisms of $mathbb{C}^2$ of the form $F(z,w)= (p(z), q(z,w))$ that extend to holomorphic endomorphisms of $mathbb{P}^2(mathbb{C})$. We prove that dynamical stability in the sense of arXiv:1403.7603 preserves hyperbolicity within such families, and give a complete classification of the hyperbolic components that are the analogue, in this setting, of the complement of the Mandelbrot set for the family $z^2 +c$. We also precisely describe the geometry of the bifurcation locus and current near the boundary of the parameter space. One of our tools is an asymptotic equidistribution property for the bifurcation current. This is established in the general setting of families of endomorphisms of $mathbb{P}^k$ and is the first equidistribution result of this kind for holomorphic dynamical systems in dimension larger than one.
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