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

ims19-06
Dzmitry Dudko, Mikhail Lyubich
Local connectivity of the Mandelbrot set at some satellite parameters of bounded type
Abstract:

We explore geometric properties of the Mandelbrot set M, and the corresponding Julia sets J_c, near the main cardioid. Namely, we establish that: a) M is locally connected at certain infinitely renormalizable parameters c of bounded satellite type, providing first examples of this kind; b) The Julia sets J_c are also locally connected and have positive area; c) M is self-similar near Siegel parameters of constant type. We approach these problems by analyzing the unstable manifold of the pacman renormalization operator constructed in [DLS] as a global transcendental family.

arXiv:1808.10425v3

Russell Lodge, Mikhail Lyubich, Sergei Merenkov, Sabyasachi Mukherjee
On Dynamical Gaskets Generated by Rational Maps, Kleinian Groups, and Schwarz Reflections
Abstract:

According to the Circle Packing Theorem, any triangulation of the Riemann sphere can be realized as a nerve of a circle packing. Reflections in the dual circles generate a Kleinian group H whose limit set is an Apollonian-like gasket ΛH. We design a surgery that relates H to a rational map g whose Julia set Jg is (non-quasiconformally) homeomorphic to ΛH. We show for a large class of triangulations, however, the groups of quasisymmetries of ΛH and Jg are isomorphic and coincide with the corresponding groups of self-homeomorphisms. Moreover, in the case of H, this group is equal to the group of Möbius symmetries of ΛH, which is the semi-direct product of H itself and the group of Möbius symmetries of the underlying circle packing. In the case of the tetrahedral triangulation (when ΛH is the classical Apollonian gasket), we give a piecewise affine model for the above actions which is quasiconformally equivalent to g and produces H by a David surgery. We also construct a mating between the group and the map coexisting in the same dynamical plane and show that it can be generated by Schwarz reflections in the deltoid and the inscribed circle.

arXiv:1912.13438

ims18-01
Araceli Bonifant, John Milnor
Group Actions, Divisors, and Plane Curves
Abstract:

After a general discussion of group actions, orbifolds, and "weak orbifolds" this note will provide elementary introductions to two basic moduli spaces over the real or complex numbers: First the moduli space of effective divisors with finite stabilizer on the projective space ℙ1 modulo the group PGL2 of projective transformations of ℙ1; and then the moduli space of effective 1-cycles with finite stabilizer on ℙ2 modulo the group PGL3 of projective transformations of ℙ2.

 arXiv:1809.05191
ims18-02
Jeffrey Brock, Christopher Leininger, Babak Modami, Kasra Rafi
Abstract:

In this paper we prove that the limit set of any Weil-Petersson geodesic ray with uniquely ergodic ending lamination is a single point in the Thurston compactification of Teichmüller space. On the other hand, we construct examples of Weil-Petersson geodesics with minimal nonuniquely ergodic ending laminations and limit set a circle in the Thurston compactification.

arXiv:1611.02197v3

ims18-03
Dzmitry Dudko, Mikhail Lyubich
Local connectivity of the Mandelbrot set at some satellite parameters of bounded type
Abstract:

We explore geometric properties of the Mandelbrot set M, and the corresponding Julia sets J_c, near the main cardioid. Namely, we establish that: a) M is locally connected at certain infinitely renormalizable parameters c of bounded satellite type, providing first examples of this kind; b) The Julia sets J_c are also locally connected and have positive area; c) M is self-similar near Siegel parameters of constant type. We approach these problems by analyzing the unstable manifold of the pacman renormalization operator constructed in [DLS] as a global transcendental family.

arXiv:1808.10425

ims18-06
Seung-Yeop Lee, Mikhail Lyubich, Nikolai G. Makarov, Sabyasachi Mukherjee
Dynamics of Schwarz reflections: the mating phenomena
Abstract:

In this paper, we initiate the exploration of a new class of anti-holomorphic dynamical systems generated by Schwarz reflection maps associated with quadrature domains. More precisely, we study Schwarz reflection with respect to a deltoid, and Schwarz reflections with respect to a cardioid and a family of circumscribing circles. We describe the dynamical planes of the maps in question, and show that in many cases, they arise as unique conformal matings of quadratic anti-holomorphic polynomials and the ideal triangle group.

 arXiv:1811.04979
ims18-07
Seung-Yeop Lee, Mikhail Lyubich, Nikolai G. Makarov, Sabyasachi Mukherjee
Schwarz reflections and the Tricorn
Abstract:

We continue our study of the family  of Schwarz reflection maps with respect to a cardioid and a circle which was started in [LLMM1]. We prove that there is a natural combinatorial bijection between the geometrically finite maps of this family and those of the basilica limb of the Tricorn, which is the connectedness locus of quadratic anti-holomorphic polynomials. We also show that every geometrically finite map in  arises as a conformal mating of a unique geometrically finite quadratic anti-holomorphic polynomial and a reflection map arising from the ideal triangle group. We then follow up with a combinatorial mating description for the "periodically repelling" maps in . Finally, we show that the locally connected topological model of the connectedness locus of  is naturally homeomorphic to such a model of the basilica limb of the Tricorn.

arXiv:1812.01573

M. Martens and B. Winkler
Instability of renormalization
Abstract:

In the theory of renormalization for classical dynamical systems, e.g. unimodal maps and critical circle maps, topological conjugacy classes are stable manifolds of renormalization. Physically more realistic systems on the other hand may exhibit instability of renormalization within a topological class. This instability gives rise to new phenomena and opens up directions of inquiry that go beyond the classical theory. In phase space it leads to the coexistence phenomenon, i.e. there are systems whose attractor has bounded geometry but which are topologically conjugate to systems whose attractor has degenerate geometry; in parameter space it causes dimensional discrepancy, i.e. a topologically full family has too few dimensions to realize all possible geometric behavior.

M. Lyubich and H. Peters
Structure of partially hyperbolic Hènon maps
Abstract:

We consider the structure of substantially dissipative complex Hènon maps admitting a dominated splitting on the Julia set. The dominated splitting assumption corresponds to the one-dimensional assumption that there are no critical points on the Julia set. Indeed, we prove the corresponding description of the Fatou set, namely that it consists of only finitely many components, each either attracting or parabolic periodic. In particular there are no rotation domains, and no wandering components. Moreover, we show that $J = J^\star$ and the dynamics on $J$ is hyperbolic away from parabolic cycles.

A. Dudko and S. Sutherland
On the Lebesgue measure of the Feigenbaum Julia set
Abstract:

We show that the Julia set of the Feigenbaum polynomial has Hausdorff dimension less than 2 (and consequently it has zero Lebesgue measure). This solves a long-standing open question.