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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.
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 |
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.
The goal of this article is to prove a rigidity result for unicritical polynomials with parabolic cycles. More precisely, we show that if two unicritical polynomials have conformally conjugate parabolic germs, then the polynomials are affinely conjugate.
| arXiv:1609.09465 |
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.
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.
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.
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.
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.
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.