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Leandro Arosio

Publications and source records attributed to Leandro Arosio.

At least 37 records · Page 2Linked to original sources
arXiv2019

Backward orbits in the unit ball

We show that, if $f\colon \mathbb{B}^q\to \mathbb{B}^q$ is a holomorphic self-map of the unit ball in $\mathbb{C}^q$ and $ζ\in \partial \mathbb{B}^q$ is a boundary repelling fixed point with dilation $λ>1$, then there exists a backward orbit converging to $ζ$ with step $\log λ$. Morever, any two backward orbits converging to the same boundary repelling fixed point stay at finite distance. As a consequence there exists a unique canonical pre-model $(\mathbb{B}^k,\ell, τ)$ associated with $ζ$ where $1\leq k\leq q$, $τ$ is a hyperbolic automorphism of $\mathbb{B}^k$, and whose image $\ell(\mathbb{B}^k)$ is precisely the set of starting points of backward orbits with bounded step converging to $ζ$. This answers questions in [8] and [3,4].

math.CV↗
arXiv2018

Chaotic holomorphic automorphisms of Stein manifolds with the volume density property

Let $X$ be a Stein manifold of dimension $n\geq 2$ satisfying the volume density property with respect to an exact holomorphic volume form. For example, $X$ could be $\mathbb{C}^n$, any connected linear algebraic group that is not reductive, the Koras-Russell cubic, or a product $Y\times\mathbb{C}$, where $Y$ is any Stein manifold with the volume density property. We prove that chaotic automorphisms are generic among volume-preserving holomorphic automorphisms of $X$. In particular, $X$ has a chaotic holomorphic automorphism. A proof for $X=\mathbb{C}^n$ may be found in work of Fornæss and Sibony. We follow their approach closely. Peters, Vivas, and Wold showed that a generic volume-preserving automorphism of $\mathbb{C}^n$, $n\geq 2$, has a hyperbolic fixed point whose stable manifold is dense in $\mathbb{C}^n$. This property can be interpreted as a kind of chaos. We generalise their theorem to a Stein manifold as above.

math.CV↗
arXiv2017

Totally real embeddings with prescribed polynomial hulls

We embed compact $C^\infty$ manifolds into $\mathbb C^n$ as totally real manifolds with prescribed polynomial hulls. As a consequence we show that any compact $C^\infty$ manifold of dimension $d$ admits a totally real embedding into $\mathbb C^{\lfloor \frac{3d}{2}\rfloor}$ with non-trivial polynomial hull without complex structure.

math.CV↗
arXiv2017

Squeezing functions and Cantor Sets

We construct "large" Cantor sets whose complements resemble the unit disk arbitrarily well from the point of view of the squeezing function, and we construct "large" Cantor sets whose complements do not resemble the unit disk from the point of view of the squeezing function. Finally we show that complements of Cantor sets arising as Julia sets of quadratic polynomials have degenerate squeezing functions, despite of having Hausdorff dimension arbitrarily close to two.

math.CV↗
arXiv2017

Simultaneous models for commuting holomorphic self-maps of the ball

We prove that a finite family of commuting holomorphic self-maps of the unit ball $\mathbb{B}^q\subset \mathbb{C}^q$ admits a simultaneous holomorphic conjugacy to a family of commuting automorphisms of a possibly lower dimensional ball, and that such conjugacy satisfies a universal property. As an application we describe when a hyperbolic and a parabolic holomorphic self-map of $\mathbb{B}^q$ can commute.

math.CV↗
arXiv2017

Dynamics of transcendental Hénon maps

The dynamics of transcendental functions in the complex plane has received a significant amount of attention. In particular much is known about the description of Fatou components. Besides the types of periodic Fatou components that can occur for polynomials, there also exist so-called Baker domains, periodic components where all orbits converge to infinity, as well as wandering domains. In trying to find analogues of these one dimensional results, it is not clear which higher dimensional transcendental maps to consider. In this paper we find inspiration from the extensive work on the dynamics of complex Hénon maps. We introduce the family of transcendental Hénon maps, and study their dynamics, emphasizing the description of Fatou components. We prove that the classification of the recurrent invariant Fatou components is similar to that of polynomial Hénon maps, and we give examples of Baker domains and wandering domains.

math.CV↗
arXiv2016

Valiron and Abel equations for holomorphic self-maps of the polydisc

We introduce a notion of hyperbolicity and parabolicity for a holomorphic self-map $f: Δ^N \to Δ^N$ of the polydisc which does not admit fixed points in $Δ^N$. We generalize to the polydisc two classical one-variable results: we solve the Valiron equation for a hyperbolic $f$ and the Abel equation for a parabolic nonzero-step $f$. This is done by studying the canonical Kobayashi hyperbolic semi-model of $f$ and by obtaining a normal form for the automorphisms of the polydisc. In the case of the Valiron equation we also describe the space of all solutions.

math.CV↗
arXiv2015

Canonical models for the forward and backward iteration of holomorphic maps

We prove the existence and the essential uniqueness of canonical models for the forward (resp. backward) iteration of a holomorphic self-map $f$ of a cocompact Kobayashi hyperbolic complex manifold, such as the ball $\mathbb{B}^q$ or the polydisc $Δ^q$. This is done performing a time-dependent conjugacy of the dynamical system $(f^n)$, obtaining in this way a non-autonomous dynamical system admitting a relatively compact forward (resp. backward) orbit, and then proving the existence of a natural complex structure on a suitable quotient of the direct limit (resp. subset of the inverse limit). As a corollary we prove the existence of a holomorphic solution with values in the upper half-plane of the Valiron equation for a holomorphic self-map of the unit ball.

math.CV↗
arXiv2015

The stable subset of a univalent self-map

We give a complete description of the stable subset (the union of all backward orbit with bounded step) and of the pre-models of a univalent self-map $f: X\to X$, where $X$ is a Kobayashi hyperbolic cocompact complex manifold, such as the ball or the polydisc in $C^q$. The result is obtained studying the complex structure of a decreasing intersection of complex manifolds, all biholomorphic to $X$.

math.CV↗
arXiv2015

Abstract basins of attraction

Abstract basins appear naturally in different areas of several complex variables. In this survey we want to describe three different topics in which they play an important role, leading to interesting open problems.

math.CV↗
arXiv2014

Canonical Models for holomorphic iteration

We construct canonical intertwining semi-models with Kobayashi hyperbolic base space for holomorphic self-maps of complex manifolds which are univalent on some absorbing cocompact hyperbolic domain. In particular, in the unit ball we solve the Valiron equation for hyperbolic univalent self-maps and for hyperbolic semigroups.

math.CV↗
arXiv2014

A survey on non-autonomous basins in several complex variables

Consider a holomorphic automorphism which acts hyperbolically on some invariant compact set. Then for every point in the compact set there exists a stable manifold, which is a complex manifold diffeomorphic to real Euclidean space. If the point is fixed, then the stable manifold is even biholomorphic to complex Euclidean space. In fact, it is known that the stable manifold of a generic point is biholomorphic to Euclidean space, and it has been conjectured that this holds for every point. In this article we survey the history of this problem, addressing both known results and the techniques used to obtain those results. Moreover, we present a list of seemingly simpler open problems and prove several new results, all pointing towards a positive answer to the conjecture discussed above.

math.CV↗
arXiv2013

Embedding univalent functions in filtering Loewner chains in higher dimension

We discuss the problem of embedding univalent functions into Loewner chains in higher dimension. In particular, we prove that a normalized univalent map of the ball in $\C^n$ whose image is a smooth strongly pseudoconvex domain is embeddable into a normalized Loewner chain (satisfying also some extra regularity properties) if and only if the closure of the image is polynomially convex.

math.CV↗
arXiv2012

Infinitesimal generators and the Loewner equation on complete hyperbolic manifolds

We characterize infinitesimal generators on complete hyperbolic complex manifolds without any regularity assumption on the Kobayashi distance. This allows to prove a general Loewner type equation with regularity of any order $d\in [1,+\infty ]$. Finally, based on these results, we focus on some open problems naturally arising.

math.CV↗
arXiv2012

Loewner equations on complete hyperbolic domains

We prove that, on a complete hyperbolic domain D\subset C^q, any Loewner PDE associated with a Herglotz vector field of the form H(z,t)=A(z)+O(|z|^2), where the eigenvalues of A have strictly negative real part, admits a solution given by a family of univalent mappings (f_t: D\to C^q) such that the union of the images f_t(D) is the whole C^q. If no real resonance occurs among the eigenvalues of A, then the family (e^{At}\circ f_t) is uniformly bounded in a neighborhood of the origin. We also give a generalization of Pommerenke's univalence criterion on complete hyperbolic domains.

math.CV↗
arXiv2012

Basins of attraction in Loewner equations

We prove that any Loewner PDE whose driving term h(z,t) vanishes at the origin, and satisfies the bunching condition r m(Dh(0,t))\geq k(Dh(0,t)) for some r\in R^+, admits a solution given by univalent mappings (f_t: B^q\to C^q). This is done by discretizing time and considering the abstract basin of attraction. If r<2, then the union of the images f_t(\B^q) of a such solution is biholomorphic to C^q.

math.CV↗
arXiv2011

An abstract approach to Loewner chains

We present a new geometric construction of Loewner chains in one and several complex variables which holds on a complete hyperbolic complex manifold M and prove that there is essentially a one-to-one correspondence between evolution families of order d and Loewner chains of the same order. As a consequence we obtain a solution for any Loewner-Kufarev PDE, given by univalent mappings (f_t) from M to a complex manifold N. The problem of finding solutions given by univalent mappings with range in C^n is reduced to investigating whether the union of the images f_t(M) is biholomorphic to a domain in C^n. We apply such results to the study of univalent mappings from the unit ball B^n to C^n.

math.CV↗