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PREPRINTS IN THIS SERIES, IN PDF FORMAT.
* Starred papers have appeared in the journal cited.

D. Dudko, M. Lyubich, and N. Selinger
Pacman renormalization and self-similarity of the Mandelbrot set near Siegel parameters
Abstract:

In the 1980s Branner and Douady discovered a surgery relating various limbs of the Mandelbrot set. We put this surgery in the framework of "Pacman Renormalization Theory" that combines features of quadratic-like and Siegel renormalizations. We show that Siegel renormalization periodic points (constructed by McMullen in the 1990s) can be promoted to pacman renormalization periodic points. Then we prove that these periodic points are hyperbolic with one-dimensional unstable manifold. As a consequence, we obtain the scaling laws for the centers of satellite components of the Mandelbrot set near the corresponding Siegel parameters.

ims17-06
Konstantin Bogdanov, Khudoyor Mamayusupov, Sabyasachi Mukherjee, Dierk Schleicher
Antiholomorphic perturbations of Weierstrass Zeta functions and Green's function on tori
Abstract:

In \cite{BeEr}, Bergweiler and Eremenko computed the number of critical points of the Green's function on a torus by investigating the dynamics of a certain family of antiholomorphic meromorphic functions on tori. They also observed that hyperbolic maps are dense in this family of meromorphic functions in a rather trivial way. In this paper, we study the parameter space of this family of meromorphic functions, which can be written as antiholomorphic perturbations of Weierstrass Zeta functions. On the one hand, we give a complete topological description of the hyperbolic components and their boundaries, and on the other hand, we show that these sets admit natural parametrizations by associated dynamical invariants. This settles a conjecture, made in \cite{LW}, on the topology of the regions in the upper half plane ℍ where the number of critical points of the Green's function remains constant.

arXiv:1612.01244v3

ims17-07
Jeffrey Brock, Christopher Leininger, Babak Modami, Kasra Rafi
Limit sets of Weil-Petersson geodesics with nonminimal ending laminations
Abstract:

In this paper we construct examples of Weil-Petersson geodesics with nonminimal ending laminations which have 1-dimensional limit sets in the Thurston compactification of Teichmüller space.

rXiv:1711.01663v2

Semi-parabolic tools for hyperbolic Henon maps and continuity of Julia sets in $\mathbb{C}^2$
Abstract:

We prove some new continuity results for the Julia sets $J$ and $J^{+}$ of the complex Hénon map $H_{c,a}(x,y)=(x^{2}+c+ay, ax)$, where $a$ and $c$ are complex parameters. We look at the parameter space of dissipative Hénon maps which have a fixed point with one eigenvalue $(1+t)\lambda$, where $\lambda$ is a root of unity and $t$ is real and small in absolute value. These maps have a semi-parabolic fixed point when $t$ is $0$, and we use the techniques that we have developed in [RT] for the semi-parabolic case to describe nearby perturbations. We show that for small nonzero $|t|$, the Hénon map is hyperbolic and has connected Julia set. We prove that the Julia sets $J$ and $J^{+}$ depend continuously on the parameters as $t\rightarrow 0$, which is a two-dimensional analogue of radial convergence from one-dimensional dynamics. Moreover, we prove that this family of Hénon maps is stable on $J$ and $J^{+}$ when $t$ is nonnegative.

T. Firsova, M. Lyubich, R. Radu, and R. Tanase
Hedgehogs for neutral dissipative germs of holomorphic diffeomorphisms of $(\mathbb{C}^{2},0)$
Abstract:

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.

M. Lyubich, R. Radu, and R. Tanase
Hedgehogs in higher dimensions and their applications
Abstract:

In this paper we study the dynamics of germs of holomorphic diffeomorphisms of $(\mathbb{C}^{n},0)$ with a fixed point at the origin with exactly one neutral eigenvalue. We prove that the map on any local center manifold of $0$ is quasiconformally conjugate to a holomorphic map and use this to transport results from one complex dimension to higher dimensions.

M. Lyubich and S. Merenkov
Quasisymmetries of the basilica and the Thompson group
Abstract:
We give a description of the group of all quasisymmetric self-maps of the Julia set of $f(z)=z^2-1$ that have orientation preserving homeomorphic extensions to the whole plane. More precisely, we prove that this group is the uniform closure of the group generated by the Thompson group of the unit circle and an inversion. Moreover, this result is quantitative in the sense that distortions of the approximating maps are uniformly controlled by the distortion of the given map.
ims16-05
Araceli Bonifant, John Milnor
On Real and Complex Cubic Curves
Abstract:

An expository description of smooth cubic curves in the real or complex projective plane.

arXiv:1603.09018v2

P. Hazard, M. Martens and C. Tresser
Infinitely Many Moduli of Stability at the Dissipative Boundary of Chaos
Abstract:

In the family of area-contracting Henon-like maps with zero topological entropy we show that there are maps with infinitely many moduli of stability. Thus one cannot find all the possible topological types for non-chaotic area-contracting Hénon-like maps in a family with finitely many parameters. A similar result, but for the chaotic maps in the family, became part of the folklore a short time after Hénon used such maps to produce what was soon conjectured to be the first non-hyperbolic strange attractors in $\mathbb{R}^2$. Our proof uses recent results about infinitely renormalisable area-contracting Hénon-like maps; it suggests that the number of parameters needed to represent all possible topological types for area-contracting Hénon-like maps whose sets of periods of their periodic orbits are finite (and in particular are equal to $\{1,\, 2,\dots,\,2^{n-1}\}$ or an initial segment of this n-tuple) increases with the number of periods. In comparison, among $C^k$-embeddings of the 2-disk with $k>0$, the maximal moduli number for non-chaotic but non area-contracting maps in the interior of the set of zero-entropy is infinite.

A. Bonifant, X. Buff and J. Milnor
Antipode Preserving Cubic Maps: the Fjord Theorem
Abstract:
This note will study a family of cubic rational maps which carry antipodal points of the Riemann sphere to antipodal points. We focus particularly on the fjords, which are part of the central hyperbolic component but stretch out to infinity. These serve to decompose the parameter plane into subsets, each of which is characterized by a corresponding rotation number.