7,433 research outputs found
Weak Hyperbolicity on Periodic Orbits for Polynomials
We prove that if the multipliers of the repelling periodic orbits of a
complex polynomial grow at least like , for some , then the Julia set of the polynomial is locally connected when it is
connected.
As a consequence for a polynomial the presence of a Cremer cycle implies the
presence of a sequence of repelling periodic orbits with "small" multipliers.
Somehow surprinsingly the proof is based in measure theorical considerations.Comment: 6 pages, Late
Statistical properties of topological Collet-Eckmann maps
We study geometric and statistical properties of complex rational maps
satisfying the Topological Collet-Eckmann Condition. We show that every such a
rational map possesses a unique conformal probability measure of minimal
exponent, and that this measure is non-atomic, ergodic and that its Hausdorff
dimension is equal to the Hausdorff dimension of the Julia set. Furthermore, we
show that there is a unique invariant probability measure that is absolutely
continuous with respect to this conformal measure, and we show that this
measure is exponentially mixing (it has exponential decay of correlations) and
that it satisfies the Central Limit Theorem.
We also show that for a complex rational map f the existence of such an
invariant measure characterizes the Topological Collet-Eckmann Condition, and
that this measure is the unique equilibrium state with potential - HD(J(f)) ln
|f'|
Nice inducing schemes and the thermodynamics of rational maps
We consider the thermodynamic formalism of a complex rational map of
degree at least two, viewed as a dynamical system acting on the Riemann sphere.
More precisely, for a real parameter we study the (non-)existence of
equilibrium states of for the potential , and the analytic
dependence on of the corresponding pressure function. We give a fairly
complete description of the thermodynamic formalism of a rational map that is
"expanding away from critical points" and that has arbitrarily small "nice
sets" with some additional properties. Our results apply in particular to
non-renormalizable polynomials without indifferent periodic points, infinitely
renormalizable quadratic polynomials with a priori bounds, real quadratic
polynomials, topological Collet-Eckmann rational maps, and to backward
contracting rational maps. As an application, for these maps we describe the
dimension spectrum of Lyapunov exponents, and of pointwise dimensions of the
measure of maximal entropy, and obtain some level-1 large deviations results.Comment: Minor adjustments in the definition of bad pull-backs of pleasant
couple
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