Note on the Equilibrium Measures of Julia sets of Exceptional Jacobi Polynomials

Exceptional sets for nonuniformly expanding maps

Given a rational map of the Riemann sphere and a subset $A$ of its Julia set, we study the $A$-exceptional set, that is, the set of points whose orbit does not accumulate at $A$. We prove that if the topological entropy of $A$ is less than the topological entropy of the full system then the $A$-exceptional set has full topological entropy. Furthermore, if the Hausdorff dimension of $A$ is smaller than the dynamical dimension of the system then the Hausdorff dimension of the $A$-exceptional set is larger than or equal to the dynamical dimension, with equality in the particular case when the dyn…

牛顿法应用于有理函数：固定点与朱利亚集

Newton's method applied to rational functions: Fixed points and Julia sets

本文研究了牛顿映射在有理函数中的固定点与朱利亚集性质，证明了特定条件下朱利亚集的连通性和局部连通性。

具有旋转对称性的有理映射的Julia集

Julia sets of rational maps with rotational symmetries

本文研究有理映射Julia集的旋转对称性，给出存在例外点时具有旋转对称性的充分条件，构造两类有限阶旋转对称性的有理映射，并证明McMullen映射的旋转对称性，且多项式与其牛顿法、切比雪夫法的旋转对称群一致。

Polynomial Diffeomorphisms of C^2: VI. Connectivity of J

Given a polynomial diffeomorphism f: C^2 -> C^2 there is a set $J_f\subset{\bf C}^2$ which we call the Julia set of f. The set $J_f\subset C^2$ plays the role of the Julia set $J\subset{\bf C}$ for a polynomial map of C. In the study of polynomial maps of C a great deal of attention has been paid to the connectivity of the Julia set. The focus of this paper is to investigate the J-connected/J-disconnected dichotomy in the case of polynomial diffeomorphisms of C^2. The Jacobian determinant of f is constant. We make the standing assumption that $|det\ Df|\le 1$ (this can always be achieved by re…

Equilibrium measures for uniformly quasiregular dynamics

We establish the existence and fundamental properties of the equilibrium measure in uniformly quasiregular dynamics. We show that a uniformly quasiregular endomorphism $f$ of degree at least 2 on a closed Riemannian manifold admits an equilibrium measure $μ_f$, which is balanced and invariant under $f$ and non-atomic, and whose support agrees with the Julia set of $f$. Furthermore we show that $f$ is strongly mixing with respect to the measure $μ_f$. We also characterize the measure $μ_f$ using an approximation property by iterated pullbacks of points under $f$ up to a set of exceptional initi…

Fekete configuration, quantitative equidistribution and wandering critical orbits in non-archimedean dynamics

Let $f$ be a rational function of degree $d>1$ on the projective line over a possibly non-archimedean algebraically closed field. A well-known process initiated by Brolin considers the pullbacks of points under iterates of $f$, and produces an important equilibrium measure. We define the asymptotic Fekete property of pullbacks of points, which means that they mirror the equilibrium measure appropriately. As application, we obtain an error estimate of equidistribution of pullbacks of points for $C^1$-test functions in terms of the proximity of wandering critical orbits to the initial points, an…