A Ruelle-McMullen formula for the volume dimension of skew products in $\mathbb C^2$
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| Format: | Preprint |
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2026
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| _version_ | 1866910043860893696 |
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| author | Bianchi, Fabrizio He, Yan Mary |
| author_facet | Bianchi, Fabrizio He, Yan Mary |
| contents | 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 $c_k(z)$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2603_06475 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A Ruelle-McMullen formula for the volume dimension of skew products in $\mathbb C^2$ Bianchi, Fabrizio He, Yan Mary Dynamical Systems Complex Variables 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 $c_k(z)$. |
| title | A Ruelle-McMullen formula for the volume dimension of skew products in $\mathbb C^2$ |
| topic | Dynamical Systems Complex Variables |
| url | https://arxiv.org/abs/2603.06475 |