arXiv subjects

Luka Boc Thaler

Publications and source records attributed to Luka Boc Thaler.

10 recordsLinked to original sources

arXiv2026

Spiral Domains and Lavaurs-Type Renormalization for Parabolic Germs of $\mathbb{C}^2$

We study the local dynamics of holomorphic germs $P:\mathbb C^2\to\mathbb C^2$ tangent to the identity whose 2-jet at the origin is $(J_0^2P)(z,w)= (z-z^2,w+w^2+bz^2)$. We prove the existence of parabolic domains for all values of the parameter $b$, showing in particular that for $b>1/4$ there are spiral domains, i.e. parabolic domains whose orbits converge to the origin without being tangent to any fixed direction. We then establish a Lavaurs-type renormalization theorem for a class of non-skew-product maps, extending earlier results known in the skew-product case. As applications, we obtain new topological invariants for such germs and construct a Fatou component with both rank-one and rank-zero limit maps. We also give an example of a polynomial self-map of $\mathbb C^3$ with an elliptic fixed point admitting a wandering domain with non-contractible limit set.

math.CV↗

arXiv2022

Dynamics of skew-products tangent to the identity

We study the local dynamics of generic skew-products tangent to the identity, i.e. maps of the form $P(z,w)=(p(z), q(z,w))$ with $dP_0=\mathrm{id}$. More precisely, we focus on maps with non-degenerate second differential at the origin; such maps have local normal form $P(z,w)=(z-z^2+O(z^3),w+w^2+bz^2+O(\|(z,w)\|^3))$. We prove the existence of parabolic domains, and prove that inside these parabolic domains the orbits converge non-tangentially if and only if $b \in (\frac{1}{4},+\infty)$. Furthermore, we prove the existence of a type of parabolic implosion, in which the renormalization limits are different from previously known cases. This has a number of consequences: under a diophantine condition on coefficients of $P$, we prove the existence of wandering domains with rank 1 limit maps. We also give explicit examples of quadratic skew-products with countably many grand orbits of wandering domains, and we give an explicit example of a skew-product map with a Fatou component exhibiting historic behaviour. Finally, we construct various topological invariants, which allow us to answer a question of Abate.

math.DS↗

arXiv2021

Reconstruction theorem for complex polynomials

Recently Takens' Reconstruction Theorem was studied in the complex analytic setting by Fornæss and Peters \cite{FP}. They studied the real orbits of complex polynomials, and proved that for non-exceptional polynomials ergodic properties such as measure theoretic entropy are carried over to the real orbits mapping. Here we show that the result from \cite{FP} also holds for exceptional polynomials, unless the Julia set is entirely contained in an invariant vertical line, in which case the entropy is $0$. In \cite{T2} Takens proved a reconstruction theorem for endomorphisms. In this case the reconstruction map is not necessarily an embedding, but the information of the reconstruction map is sufficient to recover the $2m+1$-st image of the original map. Our main result shows an analogous statement for the iteration of generic complex polynomials and the projection onto the real axis.

math.DS↗

arXiv2021

On the geometry of simply connected wandering domains

We study the geometry of simply connected wandering domains for entire functions and we prove that every bounded connected regular open set, whose closure has a connected complement, is a wandering domain of some entire function. In particular such domain can be realized as an escaping or an oscillating wandering domain. As a consequence we obtain that every Jordan curve is the boundary of a wandering Fatou component of some entire function.

math.CV↗

arXiv2020

Automorphisms of $\mathbb{C}^m$ with bounded wandering domains

We prove that the Euclidean ball can be realized as a Fatou component of a holomorphic automorphism of $\mathbb{C}^m$, in particular as the escaping and the oscillating wandering domain. Moreover, the same is true for a large class of bounded domains, namely for all bounded regular open sets $Ω\subset \mathbb{C}^m$ whose closure is polynomially convex. Our result gives in particular the first example of a bounded Fatou component with a smooth boundary in the category of holomorphic automorphisms.

math.CV↗

arXiv2020

Entire functions with prescribed singular values

We introduce a new class of entire functions $\mathcal{E}$ which consists of all $F_0\in\mathcal{O}(\mathbb{C})$ for which there exists a sequence $(F_n)\in \mathcal{O}(\mathbb{C})$ and a sequence $(λ_n)\in\mathbb{C}$ satisfying $F_n(z)=λ_{n+1}e^{F_{n+1}(z)}$ for all $n\geq 0$. This new class is closed under the composition and its is dense in the space of all non-vanishing entire functions. We prove that every closed set $V\subset \mathbb{C}$ containing the origin and at least one more point is the set of singular values of some locally univalent function in $\mathcal{E}$, hence this new class has non-trivial intersection with both the Speiser class and the Eremenko-Lyubich class of entire functions. As a consequence we provide a new proof of an old result by Heins which states that every closed set $V\subset\mathbb{C}$ is the set of singular values of some locally univalent entire function. The novelty of our construction is that these functions are obtained as a uniform limit of a sequence of entire functions, the process under which the set of singular values is not stable. Finally we show that the class $\mathcal{E}$ contains functions with an empty Fatou set and also functions whose Fatou set is non-empty.

math.CV↗

arXiv2020

Reduced dynamical systems

We consider the dynamics of complex rational maps on the Riemann sphere. We prove that, after reducing their orbits to a fixed number of positive values representing the Fubini-Study distances between finitely many initial elements of the orbit and the origin, ergodic properties of the rational map are preserved.

math.DS↗

arXiv2020

Automorphisms of $\mathbb C^2$ with parabolic cylinders

A {\sl parabolic cylinder} is an invariant, non-recurrent Fatou component $Ω$ of an automorphism $F$ of $\mathbb C^2$ satisfying: (1) The closure of the $ω$-limit set of $F$ on $Ω$ contains an isolated fixed point, (2) there exists a univalent map $Φ$ from $Ω$ into $\mathbb C^2$ conjugating $F$ to the translation $(z,w) \mapsto (z+1, w)$, and (3) every limit map of $\{F^{\circ n}\}$ on $Ω$ has one-dimensional image. In this paper we prove the existence of parabolic cylinders for an explicit class of maps, and show that examples in this class can be constructed as compositions of shears and overshears.

math.DS↗

arXiv2019

A transcendental Hénon map with an oscillating wandering Short $\mathbb{C}^2$

Short $\mathbb{C}^2$'s were constructed in [F] as attracting basins of a sequence of holomorphic automorphisms whose rate of attraction increases superexponentially. The goal of this paper is to show that such domains also arise naturally as autonomous attracting basins: we construct a transcendental Hénon map with an oscillating wandering Fatou component that is a Short $\mathbb{C}^2$. The superexponential rate of attraction is not obtained at single iterations, but along consecutive oscillations.

math.CV↗

arXiv2016

A long $\mathbb C^2$ without holomorphic functions

In this paper we construct for every integer $n>1$ a complex manifold of dimension $n$ which is exhausted by an increasing sequence of biholomorphic images of $\mathbb C^n$ (i.e., a long $\mathbb C^n$), but it does not admit any nonconstant holomorphic or plurisubharmonic functions. Furthermore, we introduce new biholomorphic invariants of a complex manifold $X$, the stable core and the strongly stable core, that are based on the long term behavior of hulls of compact sets with respect to an exhaustion of $X$. We show that every compact polynomially convex set $B\subset \mathbb C^n$ which is the closure of its interior is the strongly stable core of a long $\mathbb C^n$; in particular, biholomorphically nonequivalent sets give rise to nonequivalent long $\mathbb C^n$'s. Furthermore, for any open set $U\subset \mathbb C^n$ there exists a long $\mathbb C^n$ whose stable core is dense in $U$. It follows that for any $n>1$ there is a continuum of pairwise nonequivalent long $\mathbb C^n$'s with no nonconstant plurisubharmonic functions and no nontrivial holomorphic automorphisms. These results answer several long standing open problems.

math.CV↗