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A Ruelle-McMullen formula for the volume dimension of skew products in $\mathbb C^2$

Fabrizio Bianchi, Yan Mary He

Abstract

Ruelle gave an explicit second-order expansion at $c=0$ of the Hausdorff dimension of the Julia set of the quadratic family $f_c(z)=z^2+c$. McMullen later extended this result to polynomial perturbations of $z^d$ for arbitrary degree $d\geq 2$. In this paper we study an analogue of this problem for skew products in $\mathbb C^2$. Since holomorphic dynamical systems in higher dimensions are non-conformal, we replace the Hausdorff dimension by the \emph{volume dimension}, a dynamically defined notion we introduced in our earlier work and characterized as the zero of a natural pressure function. We consider families of holomorphic skew products of the form \[ f_t(z,w)=(z^d, w^d+t(c_1 (z) w^{d-1} +c_2(z)w^{d-2} + \cdots+c_d(z))). \] Our main result gives an explicit second-order expansion of the volume dimension of the Julia set $J(f_t)$ as $t\to0$ in terms of the coefficients $c_k(z)$.

A Ruelle-McMullen formula for the volume dimension of skew products in $\mathbb C^2$

Abstract

Ruelle gave an explicit second-order expansion at of the Hausdorff dimension of the Julia set of the quadratic family . McMullen later extended this result to polynomial perturbations of for arbitrary degree . In this paper we study an analogue of this problem for skew products in . Since holomorphic dynamical systems in higher dimensions are non-conformal, we replace the Hausdorff dimension by the \emph{volume dimension}, a dynamically defined notion we introduced in our earlier work and characterized as the zero of a natural pressure function. We consider families of holomorphic skew products of the form Our main result gives an explicit second-order expansion of the volume dimension of the Julia set as in terms of the coefficients .
Paper Structure (10 sections, 13 theorems, 49 equations)

This paper contains 10 sections, 13 theorems, 49 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Holomorphic skew products
  4. Uniform hyperbolicity and vertical expansion
  5. Conjugacies and regularity inside a hyperbolic component
  6. Proof of Theorem \ref{['thm_main_sk']}
  7. From $\ddot \delta_0$ to the variance of $\dot \phi_0$
  8. Variance of $\dot \phi_0$ as an asymptotic energy of $\partial_w v$
  9. A computation of $I(\partial_w v)$
  10. End of the proof of Theorem \ref{['thm_main_sk']}

Key Result

Theorem 1.1

Let $f_t$ be the family as in eq:form-sk. Then where $\mathop{\mathrm{Leb}}\nolimits_1$ denotes the normalized Lebesgue probability measure on $S^1$.

Theorems & Definitions (24)

  • Theorem 1.1
  • Proposition 2.1: Sester99
  • Remark 2.2
  • Proposition 2.3
  • Lemma 2.4
  • Lemma 2.5
  • Proposition 2.6
  • proof
  • Lemma 3.1
  • proof
  • ...and 14 more