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We prove that uniform hyperbolicity is invariant under topological conjugacy for dissipative polynomial automorphisms of C^2. Along the way we also show that a sufficient condition for hyperbolicity is that local stable and unstable manifolds of saddle points have uniform geometry.
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.
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.
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.
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 |
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.
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 |
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.
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.
Let f~:(S2,A~)→(S2,A~) be a Thurston map and let M(f~) be its mapping class biset: isotopy classes rel A~ of maps obtained by pre- and post-composing f~ by the mapping class group of (S2,A~). Let A⊆A~ be an f~-invariant subset, and let f:(S2,A)→(S2,A) be the induced map. We give an analogue of the Birman short exact sequence: just as the mapping class group Mod(S2,A~) is an iterated extension of Mod(S2,A) by fundamental groups of punctured spheres, M(f~) is an iterated extension of M(f) by the dynamical biset of f. Thurston equivalence of Thurston maps classically reduces to a conjugacy problem in mapping class bisets. Our short exact sequence of mapping class bisets allows us to reduce in polynomial time the conjugacy problem in M(f~) to that in M(f). In case f~ is geometric (either expanding or doubly covered by a hyperbolic torus endomorphism) we show that the dynamical biset B(f) together with a "portrait of bisets" induced by A~ is a complete conjugacy invariant of f~. Along the way, we give a complete description of bisets of (2,2,2,2)-maps as a crossed product of bisets of torus endomorphisms by the cyclic group of order 2, and we show that non-cyclic orbisphere bisets have no automorphism. We finally give explicit, efficient algorithms that solve the conjugacy and centralizer problems for bisets of expanding or torus maps.
Submitted 13 February, 2018; v1 submitted 8 February, 2018; originally announced February 2018.