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**PREPRINTS IN THIS SERIES, IN PDF FORMAT.**

* Starred papers have appeared in the journal cited.

*Bottlenecks for Weil-Petersson geodesics*

We introduce a method for constructing Weil-Petersson (WP) geodesics with certain behavior in the Teichmüller space. This allows us to study the itinerary of geodesics among the strata of the WP completion and its relation to subsurface projection coefficients of their end invariants. As an application we demonstrate the disparity between short curves in the universal curve over a WP geodesic and… ▽ More

Submitted 30 January, 2020; originally announced January 2020.

*Schwarz reflections and anti-holomorphic correspondences*

In this paper, we continue exploration of the dynamical and parameter planes of one-parameter families of Schwarz reflections that was initiated in \cite{LLMM1,LLMM2}. Namely, we consider a family of quadrature domains obtained by restricting the Chebyshev cubic polynomial to various univalent discs. Then we perform a quasiconformal surgery that turns these reflections to parabolic rational maps (which is the crucial technical ingredient of our theory). It induces a straightening map between the parameter plane of Schwarz reflections and the parabolic Tricorn. We describe various properties of this straightening highlighting the issues related to its anti-holomorphic nature. We complete the discussion by comparing our family with the classical Bullett-Penrose family of matings between groups and rational maps induced by holomorphic correspondences. More precisely, we show that the Schwarz reflections give rise to anti-holomorphic correspondences that are matings of parabolic anti-rational maps with the abstract modular group. We further illustrate our mating framework by studying the correspondence associated with the Schwarz reflection map of a deltoid.

*Local connectivity of the Mandelbrot set at some satellite parameters of bounded type*

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.

*Univalent Polynomials and Hubbard Trees*

We study rational functions f of degree d+1 such that f is univalent in the exterior unit disc, and the image of the unit circle under f has the maximal number of cusps (d+1) and double points (d−2). We introduce a bi-angled tree associated to any such f. It is proven that any bi-angled tree is realizable by such an f, and moreover, f is essentially uniquely determined by its associated bi-angled tree. This combinatorial classification is used to show that such f are in natural 1:1 correspondence with anti-holomorphic polynomials of degree d with d−1 distinct, fixed critical points (classified by their Hubbard trees).

arXiv:1908.05813 |

*On Dynamical Gaskets Generated by Rational Maps, Kleinian Groups, and Schwarz Reflections*

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.

*Group Actions, Divisors, and Plane Curves*

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 |

*Limit sets of Weil-Petersson geodesics*

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.

*Local connectivity of the Mandelbrot set at some satellite parameters of bounded type*

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.

*Dynamics of Schwarz reflections: the mating phenomena*

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 |

*Schwarz reflections and the Tricorn*

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.