Real Analyticity of Hausdorff Dimension of Julia Sets of Parabolic Polynomials

Geometric Thermodynamical Formalism and Real Analyticity for Meromorphic Functions of Finite Order

Working with well chosen Riemannian metrics and employing Nevanlinna’s theory, we make the thermodynamical formalism work for a wide class of hyperbolic meromorphic functions of finite order (including in particular exponential family, elliptic functions, cosine, tangent and the cosine–root family and also compositions of these functions with arbitrary polynomials). In particular, the existence...

On the Hausdorff dimension of Julia sets of some real polynomials

The Hausdorff Dimension of the Boundary of the Mandelbrot Set and Julia Sets

It is shown that the boundary of the Mandelbrot set M has Hausdorff dimension two and that for a generic c ∈ ∂M , the Julia set of z 7→ z + c also has Hausdorff dimension two. The proof is based on the study of the bifurcation of parabolic periodic points.

Parabolic Cantor sets

The notion of a parabolic Cantor set is introduced allowing in the definition of hyperbolic Cantor sets some fixed points to have derivatives of modulus one. Such difference in the assumptions is reflected in geometric properties of these Cantor sets. It turns out that if the Hausdorff dimension of this set is denoted by h, then its h-dimensional Hausdorff measure vanishes but the h-dimensional...

Real Analyticity of Hausdorff Dimension for Expanding Rational Semigroups

We consider the dynamics of expanding semigroups generated by finitely many rational maps on the Riemann sphere. We show that for an analytic family of such semigroups, the Bowen parameter function is real-analytic and plurisubharmonic. Combining this with a result obtained by the first author, we show that if for each semigroup of such an analytic family of expanding semigroups satisfies the o...