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Anna Miriam Benini

Publications and source records attributed to Anna Miriam Benini.

25 records · Page 2Linked to original sources
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↗
arXiv2015

Permutable entire functions and multiply connected wandering domains

Let $f$ and $g$ be permutable transcendental entire functions. We use a recent analysis of the dynamical behaviour in multiply connected wandering domains to make progress on the long standing conjecture that the Julia sets of $f$ and $g$ are equal; in particular, we show that $J(f)=J(g)$ provided that neither $f$ nor $g$ has a simply connected wandering domain in the fast escaping set.

math.DS↗
arXiv2014

Expansivity properties and rigidity for non-recurrent exponential maps

We show that an exponential map $f_c(z)=e^z+c$ whose singular value $c$ is combinatorially non-recurrent and non-escaping is uniquely determined by its combinatorics, i.e. the pattern in which its dynamic rays land together. We do this by constructing puzzles and parapuzzles in the exponential family. We also prove a theorem about hyperbolicity of the postsingular set in the case that the singular value is non-recurrent. Finally, we show that boundedness of the postsingular set implies combinatorial non-recurrence if $c$ is in the Julia set.

math.DS↗
arXiv2014

Repelling periodic points and landing of rays for post-singularly bounded exponential maps

We show that repelling periodic points are landing points of periodic rays for exponential maps whose singular value has bounded orbit. For polynomials with connected Julia sets, this is a celebrated theorem by Douady, for which we present a new proof. In both cases we also show that points in hyperbolic sets are accessible by at least one and at most finitely many rays. For exponentials this allows us to conclude that the singular value itself is accessible.

math.DS↗
arXiv2014

Singular values and bounded Siegel disks

Let $f$ be an entire transcendental function of finite order and $Δ$ be a forward invariant bounded Siegel disk for $f$ with rotation number in Herman's class $\mathcal{H}$. We show that if $f$ has two singular values with bounded orbit, then the boundary of $Δ$ contains a critical point. We also give a criterion under which the critical point in question is recurrent. We actually prove a more general theorem with less restrictive hypotheses, from which these results follow.

math.DS↗
arXiv2014

A separation theorem for entire transcendental maps

We study the distribution of periodic points for a wide class of maps, namely entire transcendental functions of finite order and with bounded set of singular values, or compositions thereof. Fix $p\in\N$ and assume that all dynamic rays which are invariant under $f^p$ land. An interior $p$-periodic point is a fixed point of $f^p$ which is not the landing point of any periodic ray invariant under $f^p$. Points belonging to attracting, Siegel or Cremer cycles are examples of interior periodic points. For functions as above we show that rays which are invariant under $f^p$, together with their landing points, separate the plane into finitely many regions, each containing exactly one interior $p-$periodic point or one parabolic immediate basin invariant under $f^p$. This result generalizes the Goldberg-Milnor Separation Theorem for polynomials, and has several corollaries. It follows, for example, that two periodic Fatou components can always be separated by a pair of periodic rays landing together; that there cannot be Cremer points on the boundary of Siegel discs; that "hidden components" of a bounded Siegel disc have to be either wandering domains or preperiodic to the Siegel disc itself; or that there are only finitely many non-repelling cycles of any given period, regardless of the number of singular values.

math.DS↗