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


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.
M. Arfeux
Reading escaping trees from Hubbard trees in $\mathcal{S}_n$
Abstract:

We prove that the parameter space of monic centered cubic polynomials with a critical point of exact period n = 4 is connected. The techniques developed for this proof work for every n and provide an interesting relation between escaping trees of DeMarco-McMullen and Hubbard trees.

P. Guarino, M. Martens, and W. de Melo
Rigidity of critical circle maps
Abstract:

We prove that any two $C^4$ critical circle maps with the same irrational rotation number and the same odd criticality are conjugate to each other by a $C^1$ circle diffeomorphism. The conjugacy is $C^{1+\alpha}$ for Lebesgue almost every rotation number.

B. Winckler and M. Martens
Physical Measures for Infinitely Renormalizable Lorenz Maps
Abstract:

A physical measure on the attractor of a system describes the statistical behavior of typical orbits. An example occurs in unimodal dynamics. Namely, all infinitely renormalizable unimodal maps have a physical measure. For Lorenz dynamics, even in the simple case of infinitely renormalizable systems, the existence of physical measures is more delicate. In this article we construct examples of infinitely renormalizable Lorenz maps which do not have a physical measure. A priori bounds on the geometry play a crucial role in (unimodal) dynamics. There are infinitely renormalizable Lorenz maps which do not have a priori bounds. This phenomenon is related to the position of the critical point of the consecutive renormalizations. The crucial technical ingredient used to obtain these examples without a physical measure, is the control of the position of these critical points.

ims15-06
Hiroyuki Inou, Sabyasachi Mukherjee
Non-landing parameter rays of the multicorns
Abstract:

It is well known that every rational parameter ray of the Mandelbrot set lands at a single parameter. We study the rational parameter rays of the multicorn ∗d, the connectedness locus of unicritical antiholomorphic polynomials of degree d, and give a complete description of their accumulation properties. One of the principal results is that the parameter rays accumulating on the boundaries of odd period (except period 1) hyperbolic components of the multicorns do not land, but accumulate on arcs of positive length consisting of parabolic parameters.
We also show the existence of undecorated real-analytic arcs on the boundaries of the multicorns, which implies that the centers of hyperbolic components do not accumulate on the entire boundary of ∗d, and the Misiurewicz parameters are not dense on the boundary of ∗d.

arXiv:1406.3428v3

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