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Feliks Przytycki

Publications and source records attributed to Feliks Przytycki.

24 records · Page 2Linked to original sources
arXiv1995

On measure and Hausdorff dimension of Julia sets for holomorphic Collet--Eckmann maps

Let $f:\bar\bold C\to\bar\bold C$ be a rational map on the Riemann sphere , such that for every $f$-critical point $c\in J$ which forward trajectory does not contain any other critical point, $|(f^n)'(f(c))|$ grows exponentially fast (Collet--Eckmann condition), there are no parabolic periodic points, and else such that Julia set is not the whole sphere. Then smooth (Riemann) measure of the Julia set is 0. For $f$ satisfying additionally Masato Tsujii's condition that the average distance of $f^n(c)$ from the set of critical points is not too small, we prove that Hausdorff dimension of Julia set is less than 2. This is the case for $f(z)=z^2+c$ with $c$ real, $0\in J$, for a positive line measure set of parameters $c$.

math.DS↗
arXiv1994

Iterations of rational functions: which hyperbolic components contain polynomials?

Let $H^d$ be the set of all rational maps of degree $d\ge 2$ on the Riemann sphere which are expanding on Julia set. We prove that if $f\in H^d$ and all or all but one critical points (or values) are in the immediate basin of attraction to an attracting fixed point then there exists a polynomial in the component $H(f)$ of $H^d$ containing $f$. If all critical points are in the immediate basin of attraction to an attracting fixed point or parabolic fixed point then $f$ restricted to Julia set is conjugate to the shift on the one-sided shift space of $d$ symbols. We give exotic examples of maps of an arbitrary degree $d$ with a non-simply connected, completely invariant basin of attraction and arbitrary number $k \ge 2$ of critical points in the basin. For such a map $f\in H^d$ with $k<d$ there is no polynomial in $H(f)$. Finally we describe a computer experiment joining an exotic example to a Newton's method (for a polynomial) rational function with a 1-parameter family of rational maps.

math.DS↗
arXiv1993

Accessability of typical points for invariant measures of positive Lyapunov exponents for iterations of holomorphic maps

We prove that if A is the basin of immediate attraction to a periodic attracting or parabolic point for a rational map f on the Riemann sphere, if $A$ is completely invariant (i.e. $f^{-1}(A)=A$), and if $μ$ is an arbitrary $f$-invariant measure with positive Lyapunov exponents on the boundary of $A$, then $μ$-almost every point $q$ in the boundary of $A$ is accessible along a curve from $A$. In fact we prove the accessability of every "good" $q$ i.e. such $q$ for which "small neighbourhoods arrive at large scale" under iteration of $f$. This generalizes Douady-Eremenko-Levin-Petersen theorem on the accessability of periodic sources.

math.DS↗
arXiv1992

Problems in holomorphic dynamics

Contents: 1. Quasiconformal Surgery and Deformations: Ben Bielefeld, Questions in quasiconformal surgery; Curt McMullen, Rational maps and Teichmüller space; John Milnor, Thurston's algorithm without critical finiteness; Mary Rees, A possible approach to a complex renormalization problem. 2. Geometry of Julia Sets: Lennart Carleson, Geometry of Julia sets; John Milnor, Problems on local connectivity. 3. Measurable Dynamics: Mikhail Lyubich, Measure and Dimension of Julia Sets; Feliks Przytycki, On invariant measures for iterations of holomorphic maps. 4. Iterates of Entire Functions: Robert Devaney, Open questions in non-rational complex dynamics; Alexandre Eremenko and Mikhail Lyubich, Wandering domains for holomorphic maps. 5. Newton's Method: Scott Sutherland, Bad polynomials for Newton's method

math.DS↗
arXiv1992

Cantor sets in the line: scaling function and the smoothness of the shift map

Consider $d$ disjoint closed subintervals of the unit interval and consider an orientation preserving expanding map which maps each of these subintervals to the whole unit interval. The set of points where all iterates of this expanding map are defined is a Cantor set. Associated to the construction of this Cantor set is the scaling function which records the infinitely deep geometry of this Cantor set. This scaling function is an invariant of $C^1$ conjugation. We solve the inverse problem posed by Dennis Sullivan: given a scaling function, determine the maximal possible smoothness of any expanding map which produces it.

math.DS↗