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Jonguk Yang

Publications and source records attributed to Jonguk Yang.

13 recordsLinked to original sources
arXiv2025

Rigidity of bounded-type Siegel polynomials

We establish rigidity for a class of higher-degree complex polynomials with irrationally indifferent dynamics. Specifically, we consider non-renormalizable (in the sense of Douady and Hubbard) polynomials of degree $d\geqslant 2$ with a Siegel disk whose rotation number is of bounded type. We call such maps atomic Siegel polynomials of bounded type. Our main results are: (A) The Julia set of every atomic Siegel polynomial of bounded type is locally connected; (B) Every atomic Siegel polynomial of bounded type is quasiconformally rigid; equivalently, its Julia set supports no invariant line fields; (C) Any two combinatorially equivalent atomic Siegel polynomials of bounded type are affinely conjugate. In particular, (C) proves the Combinatorial Rigidity Conjecture for atomic Siegel polynomials of bounded type in arbitrary degree. This extends the higher-degree rigidity theory of Avila--Kahn--Lyubich--Shen and Kozlovski--van Strien to the setting of irrationally indifferent dynamics, a setting not previously covered by Yoccoz-type rigidity results.

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arXiv2024

Quantitative Estimates on Invariant Manifolds for Surface Diffeomorphisms

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 H\'enon-like maps developed in the sequel papers.

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arXiv2024

A Priori Bounds for H\'enon-like Renormalization

We formulate and prove $\textit{a priori}$ bounds for the renormalization of H\'enon-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.

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arXiv2024

On Regular H\'enon-like Renormalization

We develop a renormalization theory of non-perturbative dissipative H\'enon-like maps with combinatorics of bounded type. The main novelty of our approach is the incorporation of Pesin theoretic ideas to the renormalization method, which enables us to control the small-scale geometry of dynamics in the higher-dimensional setting. In a prequel to this paper, it is shown that, under certain regularity conditions on the return maps, renormalizations of H\'enon-like maps have $\textit{a priori}$ bounds. The current paper is devoted to the applications of this critical estimate. First, we prove that H\'enon-like maps converge under renormalization to the same renormalization attractor as for 1D unimodal maps. Second, we show that the necessary and sufficient conditions for renormalization convergence are finite-time checkable. Lastly, we show that every infinitely renormalizable H\'enon-like map is $\textit{regularly unicritical}$: there exists a unique orbit of tangencies between strong-stable and center manifolds, and outside a slow-exponentially shrinking neighborhood of this orbit, the dynamics behaves as a uniformly partially hyperbolic system.

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arXiv2024

Rigidity of bounded type cubic Siegel polynomials

We prove that if two non-renormalizable cubic Siegel polynomials with bounded type rotation numbers are combinatorially equivalent, then they are also conformally equivalent. As a consequence, we show that in the one-parameter slice of cubic Siegel polynomials considered by Zakeri [Za2], the locus of non-renormalizable maps is homeomorphic to a double-copy of a quadratic Siegel filled Julia set (minus the Siegel disk) glued along the Siegel boundary. This verifies the the conjecture of Blokh-Oversteegen-Ptacek-Timorin [BlOvPtTi] for bounded type rotation numbers.

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arXiv2024

Renormalization of Unicritical Diffeomorphisms of the Disk

We introduce a class of infinitely renormalizable, unicritical diffeomorphisms of the disk (with a non-degenerate "critical point"). In this class of dynamical systems, we show that under renormalization, maps eventually become Hénon-like, and then converge super-exponentially fast to the space of one-dimensional unimodal maps. We also completely characterize the local geometry of every stable and center manifolds that exist in these systems. The theory is based upon a quantitative reformulation of the Oseledets-Pesin theory yielding a unicritical structure of the maps in question comprising regular Pesin boxes co-existing with "critical tunnels" and "valuable crescents". In forthcoming notes we will show that infinitely renormalizable perturbative Hénon-like maps of bounded type belong to our class.

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arXiv2023

Local Connectivity of Polynomial Julia sets at Bounded Type Siegel Boundaries

Consider a polynomial $f$ of degree $d \geq 2$ that has a Siegel disk $Δ_f$ with a rotation number of bounded type. We prove that there does not exist a hedgehog containing $Δ_f$. Moreover, if the Julia set $J_f$ of $f$ is connected, then it is locally connected at the Siegel boundary $\partial Δ_f$.

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arXiv2020

Elastic Graphs for Main Molecule Matings

Recent work of Dylan Thurston gives a condition for when a post-critically finite branched self-cover of the sphere is equivalent to a rational map. We apply D. Thurston's positive criterion for rationality to give a new proof of a theorem of Rees, Shishikura, and Tan about the mateability of quadratic polynomials when one polynomial is in the main molecule. These methods may be a step in understanding the mateability of higher degree post-critically finite polynomials and demonstrate how to apply the positive criterion to classical problems.

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arXiv2019

Structural Instability of Semi-Siegel Hénon maps

We show that the dynamics of sufficiently dissipative semi-Siegel complex Hénon maps with golden-mean rotation number is not $J$-stable in a very strong sense. By the work of Dujardin and Lyubich, this implies that the Newhouse phenomenon occurs for a dense $G_δ$ set of parameters in this family. Another consequence is that the Julia sets of such maps are disconnected for a dense set of parameters.

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arXiv2017

Renormalization in the Golden-Mean Semi-Siegel Hénon Family: Universality and Non-Rigidity

It was recently shown by Gaidashev and Yampolsky that appropriately defined renormalizations of a sufficiently dissipative golden-mean semi-Siegel Hénon map converge super-exponentially fast to a one-dimensional renormalization fixed point. In this paper, we show that the asymptotic two-dimensional form of these renormalizations is universal, and is parameterized by the average Jacobian. This is similar to the limit behavior of period-doubling renormalization in the Hénon family considered by de Carvalho, Lyubich and Martens. As an application of our result, we prove that the boundary of the golden-mean Siegel disk of a dissipative Hénon map is non-rigid.

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arXiv2014

Mating the Basilica with a Siegel Disk

Consider a quadratic polynomial with a fixed Siegel disc of bounded type. Using an adaptation of complex a priori bounds for critical circle maps, we prove that this Siegel polynomial is conformally mateable with the basilica polynomial.

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