Dynamical Systems Seminar: Hasina Akter (UNT): Real Analyticity of Hausdorff Dimension of Julia Sets of Parabolic Polynomials $f_{\lambda}(z)=z(1-z-\lambda z^2 )$, GAB 461

Event Information
Dynamical Systems Seminar
Event Date: 
Repeats every week until Thu May 03 2012 .
04-12-2012 3:00 PM
Event Location: 
GAB 461

Abstract: Let $D_0$ denote the set of all parameters $\lambda\in\C\setminus\{0\}$ for which the cubic polynomial $f_\lambda$ is parabolic and has no parabolic or finite attracting periodic cycles other than $0$. We prove that $D_0$ contains a deleted neighborhood of the origin $0$. Our main result is that the function $D_0\ni\lambda\mapsto\text{HD}(J(f_\lambda))\in\R$ is real-analytic. This function ascribes to the polynomial $f_\lambda$ the Hausdorff dimension of its Julia set $J(f_\lambda)$. The theory of parabolic and hyperbolic graph directed Markov systems with infinite number of edges is used in the proofs.