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We formulate and prove a priori bounds for the renormalization of Hénon-like maps (under certain regularity assumptions). This provides a certain uniform control on the small-scale geometry of the dynamics, and ensures pre-compactness of the renormalization sequence. In a sequel to this paper, a priori bounds are used in the proof of the main results, including renormalization convergence, finite-time checkability of the required regularity conditions and regular unicriticality of the dynamics.
Submitted 20 November, 2024; originally announced November 2024.
We carry out a detailed quantitative analysis on the geometry of invariant manifolds for smooth dissipative systems in dimension two. We begin by quantifying the regularity of any orbit (finite or infinite) in the phase space with a set of explicit inequalities. Then we relate this directly to the quasi-linearization of the local dynamics on regular neighborhoods of this orbit. The parameters of regularity explicitly determine the sizes of the regular neighborhoods and the smooth norms of the corresponding regular charts. As a corollary, we establish the existence of smooth stable and center manifolds with uniformly bounded geometries for regular orbits independently of any pre-existing invariant measure. This provides us with the technical background for the renormalization theory of Hénon-like maps developed in the sequel papers.
Submitted 20 November, 2024; originally announced November 2024.
We consider certain analytic correspondences on a Riemann surface, and show that they admit a weak form of expansion. In terms of their algebraic encoding by bisets, this translates to contraction of group elements along sequences arising from iterated lifting. As an application, we show that for every non-exceptional rational map on P1 with 4 post-critical points, there is a finite collection of isotopy classes of curves into which every curve eventually lands under iterated lifting.
Submitted 22 July, 2024; originally announced July 2024.
We study the parameter space Sp for cubic polynomial maps with a marked critical point of period p. We will outline a fairly complete theory as to how the dynamics of the map F changes as we move around the parameter space Sp. For every escape region E⊂Sp, every parameter ray in E with rational parameter angle lands at some uniquely defined point in the boundary ∂E. This landing point is necessarily either a parabolic map or a Misiurewicz map. The relationship between parameter rays and dynamic rays is formalized by the period q tessellation of Sp, where maps in the same face of this tessellation always have the same period q orbit portrait.
Submitted 11 March, 2025; originally announced March 2025.
For a post-critically finite hyperbolic rational map f, we show that the Julia set Jf has Ahlfors-regular conformal dimension one if and only if f is a crochet map, i.e., there is an f-invariant graph G containing the post-critical set such that f|G has topological entropy zero. We use finite subdivision rules to obtain graph virtual endomorphisms, which are 1-dimensional simplifications of post-critically finite rational maps, and approximate the asymptotic conformal energies of the graph virtual endomorphisms to estimate the Ahlfors-regular conformal dimensions. In particular, we develop an idea of reducing finite subdivision rules and prove the monotonicity of asymptotic conformal energies under the decomposition of rational maps.
Let p be a prime number, let g(x)=xp2+pr+2xp2+1 with r∈ℤ≥0, and let ϕ(x)=x+O(x2) be the Böttcher coordinate satisfying ϕ(g(x))=ϕ(x)p2. Salerno and Silverman conjectured that the radius of convergence of ϕ−1(x) in ℂp is p−p−r/(p−1). In this article, we confirm that this conjecture is true by showing that it is a special case of our more general result.
In this paper, we study the dynamics of a general class of antiholomorphic correspondences; i.e., multi-valued maps with antiholomorphic local branches, on the Riemann sphere. Such correspondences are closely related to a class of single-valued antiholomorphic maps in one complex variable; namely, Schwarz reflection maps of simply connected quadrature domains. Using this connection, we prove that matings of all parabolic antiholomorphic rational maps with connected Julia sets (of arbitrary degree) and antiholomorphic analogues of Hecke groups can be realized as such correspondences. We also draw the same conclusion when parabolic maps are replaced with critically non-recurrent antiholomorphic polynomials with connected Julia sets.
| arXiv:2303.02459 |
Abstract: We completely characterise the bounded sets that arise as components of the Fatou and Julia sets of meromorphic functions. On the one hand, we prove that a bounded domain is a Fatou component of some meromorphic function if and only if it is regular. On the other hand, we prove that a planar continuum is a Julia component of some meromorphic function if and only if it has empty interior. We do so by constructing meromorphic functions with wandering continua using approximation theory.
In this paper, we use iterations of skinning maps on Teichmüller spaces to study circle packings. This allows us to develop a renormalization theory for circle packings whose nerves satisfy certain subdivision rules. We characterize when the skinning map has bounded image. Under the corresponding condition, we prove that the renormalization operator is uniformly contracting. This allows us to give complete answers for the existence and moduli problems for such circle packings. The exponential contraction has many consequences. In particular, we prove that homeomorphisms between any two such circle packings are asymptotically conformal.
We prove {\em a priori} bounds for Feigenbaum quadratic polynomials, i.e., infinitely renormalizable polynomials fc:z↦z2+c of bounded type. It implies local connectivity of the corresponding Julia sets J(fc) and MLC (local connectivity of the Mandelbrot set $\Mandel$) at the corresponding parameters c. It also yields the scaling Universality, dynamical and parameter, for the corresponding combinatorics. The MLC Conjecture was open for the most classical period-doubling Feigenbaum parameter as well as for the complex tripling renormalizations. Universality for the latter was conjectured by Goldberg-Khanin-Sinai in the early 1980s.