Hénon maps: a list of open problems
We propose a set of questions on the dynamics of Hénon maps from the real, complex, algebraic and arithmetic points of view.
math.DS↗
arXiv subjects
Tanya Firsova
Publications and source records attributed to Tanya Firsova.
Critical locus for Hénon maps in an HOV region
We prove that the characterization of the critical locus for complex Hénon maps that are small perturbations of quadratic polynomials with disconnected Julia sets given by Firsova holds in a much larger HOV-like region from the complex horseshoe locus. The techniques of this paper are non-perturbative.
math.DS↗
On deformation spaces of quadratic rational maps
We study the group of self-equivalences of a partially postcritically finite branched cover and answer a question of Adam Epstein about contractibility of certain deformation spaces of rational maps.
math.DS↗
Accumulation set of critical points of the multipliers in the quadratic family
A parameter $c_0\in\mathbb C$ in the family of quadratic polynomials $f_c(z)=z^2+c$ is a critical point of a period $n$ multiplier, if the map $f_{c_0}$ has a periodic orbit of period $n$, whose multiplier, viewed as a locally analytic function of $c$, has a vanishing derivative at $c=c_0$. We study the accumulation set $\mathcal X$ of the critical points of the multipliers, as $n\to\infty$. This study complements the equidistribution result for the critical points of the multipliers that was previously obtained by the authors. In particular, in the current paper we prove that the accumulation set $\mathcal X$ is bounded, path connected and contains the Mandelbrot set as a proper subset. We also provide a necessary and sufficient condition for a parameter outside of the Mandelbrot set to be contained in the accumulation set $\mathcal X$ and show that this condition is satisfied for an open set of parameters. Our condition is similar in flavor to one of the conditions that define the Mandelbrot set. As an application, we get that the function that sends $c$ to the Hausdorff dimension of $f_c$, does not have critical points outside of the accumulation set $\mathcal X$.
math.DS↗
Equidistribution of critical points of the multipliers in the quadratic family
A parameter $c_0\in\mathbb C$ in the family of quadratic polynomials $f_c(z)=z^2+c$ is a critical point of a period $n$ multiplier, if the map $f_{c_0}$ has a periodic orbit of period $n$, whose multiplier, viewed as a locally analytic function of $c$, has a vanishing derivative at $c=c_0$. We prove that all critical points of period $n$ multipliers equidistribute on the boundary of the Mandelbrot set, as $n\to\infty$.
math.DS↗
Hedgehogs for neutral dissipative germs of holomorphic diffeomorphisms of $(\mathbb{C}^{2},0)$
We prove the existence of hedgehogs for germs of complex analytic diffeomorphisms of $(\mathbb{C}^{2},0)$ with a semi-neutral fixed point at the origin, using topological techniques. This approach also provides an alternative proof of a theorem of Pérez-Marco on the existence of hedgehogs for germs of univalent holomorphic maps of $(\mathbb{C},0)$ with a neutral fixed point.
math.DS↗
Geometric proof of the $λ$-lemma
We give a geometric approach to the proof of the $λ$-lemma. In particular, we point out the role pseudoconvexity plays in the proof.
math.CV↗
$λ$-Lemma for families of Riemann surfaces and the critical loci of complex Hénon maps
We prove a version of the classical $λ$-lemma for holomorphic families of Riemann surfaces. We then use it to show that critical loci for complex Hénon maps that are small perturbations of quadratic polynomials with Cantor Julia sets are all quasiconformally equivalent.
math.DS↗
Topology of generic foliations on Stein manifolds: structure of leaves and Kupka-Smale property
We study topology of leaves of 1-dimensional singular holomorphic foliations of Stein manifolds. We prove that for a generic foliation all leaves, except for at most countably many, are contractible, the rest are topological cylinders. We show that a generic foliation is complex Kupka-Smale.
math.DS↗
The Critical Locus for Complex Hénon Maps
We give a topological model of the critical locus for complex Hénon maps that are perturbations of the quadratic polynomial with disconnected Julia set.
math.DS↗