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Fabrizio Bianchi

Publications and source records attributed to Fabrizio Bianchi.

At least 19 recordsLinked to original sources
arXiv2026

Joining rigidity for rational maps

We initiate a joining rigidity theory for rational maps on the Riemann sphere $\mathbb P^1=\mathbb P^1(\mathbb C)$. Let $f_1,f_2\colon\mathbb P^1\to\mathbb P^1$ be rational maps of degree at least $2$, and $μ_1,μ_2$ their respective measures of maximal entropy, whose supports are the Julia sets $J(f_1)$ and $J(f_2)$. We study ergodic joinings of the systems $(J(f_1),f_1,μ_1)$ and $(J(f_2),f_2,μ_2)$, namely ergodic probability measures on $J(f_1)\times J(f_2)$ which are invariant under $f_1\times f_2$ and whose marginals are $μ_1$ and $μ_2$. Our main theorem shows that a positive-mass local holomorphic relation forces algebraic rigidity. More precisely, if the joining charges the graph of a local biholomorphism, then that local relation globalizes to an invariant algebraic curve and yields either a finite cycle of rational graph or transpose-graph relations, or a genuinely multi-valued invariant algebraic correspondence. If no local biholomorphic graph has positive joining measure, then the joining generates a compact non-discrete family of local holomorphic relations. The proof introduces normalized inverse branch transfer maps and studies their cluster limits. Starting from a local biholomorphic graph of positive joining measure, recurrence and contraction of inverse branches produce recurrent local intertwining relations. These are promoted to an algebraic relation by a local-to-global rigidity argument in the non-Lattès case and by affine uniformization in the Lattès case. In the absence of any positive-mass local biholomorphic graph, the cluster family must be infinite, and its non-discrete closure gives the second alternative.

math.DS↗
arXiv2026

A thermodynamic path metric for complex Hénon maps

We construct a Hermitian covariance form on hyperbolic components in parameter spaces of complex Hénon maps, associated to the full complex unstable derivative cocycle. The form measures infinitesimal variations in the marked complex unstable multiplier spectrum. Using a recent multiplier rigidity theorem by Cantat--Dujardin, we prove that it induces a distance on every hyperbolic component. Motivated by Sullivan's dictionary and by the thermodynamic interpretation of the Weil--Petersson metric, our result gives a first higher-dimensional holomorphic-dynamical counterpart of pressure-type metric structures. On the other hand, the construction differs from the one-dimensional theory in an essential way: it replaces the real geometric potential measuring unstable expansion by the full complex unstable derivative cocycle. This also suggests a complex derivative cocycle counterpart to pressure-type metric structures in Teichmüller theory and Anosov representation theory.

math.DS↗
arXiv2026

On the support of measures of large entropy for automorphisms of Kähler manifolds

Let $f$ be a holomorphic automorphism of a compact Kähler manifold $X$ with simple action on cohomology. We show that every ergodic measure with sufficiently large entropy is supported on the Julia set of $f$. In particular, when $X$ is a surface, any ergodic measure with positive entropy is supported on the Julia set. The proof relies on quantitative estimates for the speed of convergence towards the Green currents of $f$, with respect to a suitable norm on an adapted functional space of non-necessarily closed currents.

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arXiv2026

Non-autonomous parabolic implosion

We study parabolic implosion in a general non-autonomous setting. Let $f(w)=w+w^2+O(w^3)$ be a holomorphic germ tangent to the identity. We consider the iteration of non-autonomous perturbations of the form \[ w_{j+1}=f(w_j)+\varepsilon_{j,n}^2. \] We show that, when the $\varepsilon_{j,n}^2$'s satisfy a Lavaurs-type condition, the element $w_n$ can be described by means of a suitable Lavaurs map $L_{u_n}$, whose phase $u_n$ is an explicit function of the perturbation parameters. In particular, whenever $u_n\to u\in \mathbb C$, the non-autonomous dynamics converges locally uniformly on compact subsets of the parabolic basin to the corresponding Lavaurs map $L_u$. Our study provides a general description of additive non-autonomous parabolic implosion and yields several deterministic and random convergence results as corollaries, as well as a unified proof of several previous results. As an application, we also obtain strong discontinuity results for the Julia sets of fibered holomorphic endomorphisms of $\mathbb P^2(\mathbb C)$.

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arXiv2026

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

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 $c_k(z)$.

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arXiv2025

Manhattan curves in complex dynamics and asymptotic correlation of multiplier spectra

The Manhattan curve for a pair of hyperbolic structures (possibly with cusps) on a given surface is a geometric object that encodes the growth rate of lengths of closed geodesics with respect to the two different hyperbolic metrics. It has been extensively studied as a way to understand geodesics on surfaces, the thermodynamic formalism of the geodesic flows and comparison of hyperbolic metrics. Via Sullivan's dictionary, in this paper, we define and study the Manhattan curve for a pair of hyperbolic rational maps on $\mathbb C\mathbb P^1$, and more generally of holomorphic endomorphisms of $\mathbb{C}\mathbb{P}^k$. We discuss several counting results for the multiplier spectrum and show that the Manhattan curve for two holomorphic endomorphisms is related to the correlation number of their multiplier spectra.

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arXiv2025

Analyticity of the Hausdorff dimension and metric structures on Misiurewicz families of polynomials

Consider a holomorphic family $(f_λ)_{λ\in Λ}$ of polynomial maps on $\mathbb C$ with the property that a critical point of $f_λ$ is persistently preperiodic to a repelling periodic point of $f_λ$. Let $Ω$ be a bounded stable component of $Λ$ with the property that, for all $λ\in Ω$, all the other critical points of $f_λ$ belong to attracting basins. In this paper, we introduce a dynamically meaningful geometry on $Ω$ by constructing a natural path metric on $Ω$ coming from a 2-form $\langle \cdot, \cdot \rangle_G$. Our construction uses thermodynamic formalism. A key ingredient is the spectral gap of adapted transfer operators on suitable Banach spaces, which also implies the analyticity of $\langle \cdot, \cdot \rangle_G$ on the unit tangent bundle of $Ω$. As part of our construction, we recover a result of Skorulski and Urbański stating that the Hausdorff dimension of the Julia set of $f_λ$ varies analytically over $Ω$.

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arXiv2024

Pressure path metrics on parabolic families of polynomials

Let $Λ$ be a subfamily of the moduli space of degree $D\ge2$ polynomials defined by a finite number of parabolic relations. Let $Ω$ be a bounded stable component of $Λ$ with the property that all critical points are attracted by either the persistent parabolic cycles or by attracting cycles in $\mathbb C$. We construct a positive semi-definite pressure form on $Ω$ and show that it defines a path metric on $Ω$. This provides a counterpart in complex dynamics of the pressure metric on cusped Hitchin components recently studied by Kao and Bray-Canary-Kao-Martone.

math.DS↗
arXiv2024

On the support of measures of large entropy for polynomial-like maps

Let $f$ be a polynomial-like map with dominant topological degree $d_t\geq 2$ and let $d_{k-1}<d_t$ be its dynamical degree of order $k-1$. We show that the support of every ergodic measure whose measure-theoretic entropy is strictly larger than $\log \sqrt{d_{k-1} d_t}$ is supported on the Julia set, i.e., the support of the unique measure of maximal entropy $μ$. The proof is based on the exponential speed of convergence of the measures $d_t^{-n}(f^n)^*δ_a$ towards $μ$, which is valid for a generic point $a$ and with a controlled error bound depending on $a$. Our proof also gives a new proof of the same statement in the setting of endomorphisms of $\mathbb P^k(\mathbb C)$ - a result due to de Thélin and Dinh - which does not rely on the existence of a Green current.

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arXiv2024

Hölder continuity and laminarity of the Green currents for Hénon-like maps

Under a natural assumption on the dynamical degrees, we prove that the Green currents associated to any Hénon-like map in any dimension have Hölder continuous super-potentials, i.e., give Hölder continuous linear functionals on suitable spaces of forms and currents. As a consequence, the unique measure of maximal entropy is the Monge-Ampère of a Hölder continuous plurisubharmonic function and has strictly positive Hausdorff dimension. Under the same assumptions, we also prove that the Green currents are woven. When they are of bidegree $(1,1)$, they are laminar. In particular, our results generalize results known until now only in algebraic settings, or in dimension 2.

math.CV↗
arXiv2023

A Mañé-Manning formula for expanding measures for endomorphisms of $\mathbb P^k$

Let $k \ge 1$ be an integer and $f$ a holomorphic endomorphism of $\mathbb P^k (\mathbb C)$ of algebraic degree $d\geq 2$. We introduce a volume dimension for ergodic $f$-invariant probability measures with strictly positive Lyapunov exponents. In particular, this class of measures includes all ergodic measures whose measure-theoretic entropy is strictly larger than $(k-1)\log d$, a natural generalization of the class of measures of positive measure-theoretic entropy in dimension 1. The volume dimension is equivalent to the Hausdorff dimension when $k=1$, but depends on the dynamics of $f$ to incorporate the possible failure of Koebe's theorem and the non-conformality of holomorphic endomorphisms for $k\geq 2$. If $ν$ is an ergodic $f$-invariant probability measure with strictly positive Lyapunov exponents, we prove a generalization of the Mañé-Manning formula relating the volume dimension, the measure-theoretic entropy, and the sum of the Lyapunov exponents of $ν$. As a consequence, we give a characterization of the first zero of a natural pressure function for such expanding measures in terms of their volume dimensions. For hyperbolic maps, such zero also coincides with the volume dimension of the Julia set, and with the exponent of a natural (volume-)conformal measure. This generalizes results by Denker-Urbański and McMullen in dimension 1 to any dimension $k\geq 1$. Our methods mainly rely on a theorem by Berteloot-Dupont-Molino, which gives a precise control on the distortion of inverse branches of endomorphisms along generic inverse orbits with respect to measures with strictly positive Lyapunov exponents.

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arXiv2023

Holomorphic motions of weighted periodic points

We study the holomorphic motions of repelling periodic points in stable families of endomorphisms of $\mathbb P^k (\mathbb C)$. In particular, we establish an asymptotic equidistribution of the graphs associated to such periodic points with respect to natural measures in the space of all holomorphic motions of points in the Julia sets.

math.CV↗
arXiv2023

Strong probabilistic stability in holomorphic families of endomorphisms of $\mathbb{P}^k(\mathbb{C})$ and polynomial-like maps

We prove that, in stable families of endomorphisms of $\mathbb{P}^k(\mathbb{C})$, all invariant measures whose measure-theoretic entropy is strictly larger than $(k-1)\log d$ at a given parameter can be followed holomorphically with the parameter in all the parameter space. As a consequence, almost all points (with respect to any such measure at any parameter) in the Julia set can be followed holomorphically without intersections. This generalizes previous results by Berteloot, Dupont, and the first author for the measure of maximal entropy, and provides a parallel in this setting to the probabilistic stability of Hénon maps by Berger-Dujardin-Lyubich. Our proof relies both on techniques from the theory of stability/bifurcation in any dimension and on an explicit lower bound for the Lyapunov exponents for an ergodic measure in terms of its measure-theoretic entropy, due to de Thélin and Dupont. A local version of our result holds also for all measures supported on the Julia set with just strictly positive Lyapunov exponents and not charging the post-critical set. Analogous results hold in families of polynomial-like maps of large topological degree. In this case, as part of our proof, we also give a sufficient condition for the positivity of the Lyapunov exponents of an ergodic measure for a polynomial-like map in any dimension in term of its measure-theoretic entropy, generalizing to this setting the analogous result by de Thélin and Dupont valid on $\mathbb{P}^k(\mathbb{C})$.

math.DS↗
arXiv2023

Monotonicity of dynamical degrees for H{é}non-like and polynomial-like maps

We prove that, for every invertible horizontal-like map (i.e., H{é}non-like map) in any dimension, the sequence of the dynamical degrees is increasing until that of maximal value, which is the main dynamical degree, and decreasing after that. Similarly, for polynomial-like maps in any dimension, the sequence of dynamical degrees is increasing until the last one, which is the topological degree. This is the first time that such a property is proved outside of the algebraic setting. Our proof is based on the construction of a suitable deformation for positive closed currents, which relies on tools from pluripotential theory and the solution of the $d$, $\bar \partial$, and $dd^c$ equations on convex domains.

math.CV↗
arXiv2023

Exponential mixing of all orders and CLT for automorphisms of compact K{ä}hler manifolds

We consider the unique measure of maximal entropy of an automorphism of a compact K{ä}hler manifold with simple action on cohomology. We show that it is exponentially mixing of all orders with respect to H{ö}lder observables. It follows that the Central Limit Theorem (CLT) holds for these observables. In particular, our result applies to all automorphisms of compact K{ä}hler surfaces with positive entropy.

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