Transversality family of expanding rational semigroups

Abstract

We study finitely generated expanding semigroups of rational maps with overlaps on the Riemann sphere. We show that if a d-parameter family of such semigroups satisfies the transversality condition, then for almost every parameter value the Hausdorff dimension of the Julia set is the minimum of 2 and the zero of the pressure function. Moreover, the Hausdorff dimension of the exceptional set of parameters is estimated. We also show that if the zero of the pressure function is greater than 2, then typically the 2-dimensional Lebesgue measure of the Julia set is positive. Some sufficient conditions for a family to satisfy the transversality conditions are given. We give non-trivial examples of families of semigroups of non-linear polynomials with the transversality condition for which the Hausdorff dimension of the Julia set is typically equal to the zero of the pressure function and is less than 2. We also show that a family of small perturbations of the Sierpinski gasket system satisfies that for a typical parameter value, the Hausdorff dimension of the Julia set (limit set) is equal to the zero of the pressure function, which is equal to the similarity dimension. Combining the arguments on the transversality condition, thermodynamical formalisms and potential theory, we show that for each a ∈ C with |a| = 0, 1, the family of small perturbations of the semigroup generated by {z 2 , az 2 } satisfies that for a typical parameter value, the 2-dimensional Lebesgue measure of the Julia set is positive.

Bowen parameter and Hausdorff dimension for expanding rational semigroups

“…Note that combining these estimates of Bowen's parameter and the "transversal family" type arguments, we will show that we have a large amount of expanding 2-generator polynomial semigroups G such that the Julia set of G has positive 2-dimensional Lebesgue measure ( [33]).…”

Dynamics of postcritically bounded polynomial semigroups II: fiberwise dynamics and the Julia sets

“…[17] is a very nice (and short) article for an introduction to the dynamics of rational semigroups. For other research on rational semigroups, see [37,18,19,35,36], and [21]- [33].…”

Negativity of Lyapunov Exponents and Convergence of Generic Random Polynomial Dynamical Systems and Random Relaxed Newton's Methods

“…The first study of dynamics of rational semigroups was conducted by A. Hinkkanen and G. J. Martin ( [13]), who were interested in the role of the dynamics of polynomial semigroups (i.e., semigroups of non-constant polynomial maps) while studying various one-complex-dimensional moduli spaces for discrete groups, and by F. Ren's group ( [12]), who studied such semigroups from the perspective of random dynamical systems. For recent work on the dynamics of rational semigroups, see the author's papers [30]- [39], and [28,40,41].…”