arXiv subjects

Insung Park

Publications and source records attributed to Insung Park.

9 recordsLinked to original sources

arXiv2026

Puncture-Forgetting Maps for Measured Foliations and Applications in Teichm\"uller Space and Complex Dynamics

We introduce puncture-forgetting maps for measured foliations and investigate their relations with mapping class groups, Teichm\"uller spaces, and extremal length. To this end, we develop the notions of cube complexes of pre-homotopic multicurves and tree coordinate systems on CAT(0) cube complexes. As an application to the dynamics of post-critically finite rational maps on the Riemann sphere, we obtain a partial result toward the finite curve attractor conjecture posed by Kevin Pilgrim. We also apply these methods to uncover a relation between horospheres and geodesic flows in the universal curve over Teichm\"uller space.

math.DS↗

arXiv2025

Real Rational Surface Automorphisms : Positivity and Linearity

We study the real dynamics of a family of rational surface automorphisms obtained from quadratic birational maps of $\pcc$ that preserve a cuspidal cubic and whose critical orbits have lengths $(1,m,n)$ with $1+m+n\ge 10$. Passing to the real locus and cutting along the invariant cubic, we obtain a diffeomorphism of an orientable surface whose fundamental group is free. Our key device is a finitely generated invariant, positive semigroup $S_{m,n}$ in the fundamental group on which an iterate of induced action acts by concatenation without cancellation. This positivity yields a nonnegative primitive transition matrix, so Perron-Frobenius theory supplies an explicit exponential growth rate $\lambda>1$ for the induced action on the fundamental group. Consequently, the real map has positive topological entropy. We package the combinatorics of the generators in a ``Core-Tail Induction Principle," which allows us to treat simultaneously seven orbit-data families with only finite base checks. Finally, using Bestvina-Handel and the Dehn-Nielsen-Baer correspondence, we show that the induced outer automorphism with $m+n$ odd is realized by a pseudo-Anosov homeomorphism of the cut surface.

math.DS↗

arXiv2025

Analytic Theory on the Space of Blaschke Products: Simultaneous Uniformization and Pressure Metric

In this paper, we study complex analytic aspects of the moduli space $\Bcal_d^{fm}$ of degree $d\ge2$ fixed-point-marked Blaschke products. We define a complex structure on $\Bcal_d^{fm}$ and prove the simultaneous uniformization theorem for fixed-point-marked quasi-Blaschke products. As an application, we show that the pressure semi-norms on the space of Blaschke products are non-degenerate outside of the super-attracting locus $\mathcal{SA}^{fm}_d$, which is a codimension-1 subspace of $\Bcal^{fm}_d$.

math.DS↗

arXiv2024

Pressure metrics in geometry and dynamics

In this article, we first provide a survey of pressure metrics on various deformation spaces in geometry, topology, and dynamics. Then we discuss pressure semi-norms and their degeneracy loci in the space of quasi-Blaschke products

math.DS↗

arXiv2022

Julia sets with Ahlfors-regular conformal dimension one

For a post-critically finite hyperbolic rational map $f$, we show that its Julia set $\mathcal{J}_f$ has Ahlfors-regular conformal dimension one if and only if $f$ is a crochet map, i.e., there is an $f$-invariant connected 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 models of post-critically finite rational maps, and we approximate the asymptotic conformal energies of graph virtual endomorphisms to estimate the Ahlfors-regular conformal dimensions of Julia sets. To prove the main theorem, we also establish the monotonicity of asymptotic conformal energies under the decomposition of rational maps by invariant multicurves.

math.DS↗

arXiv2022

Polynomials with core entropy zero

This paper studies polynomials with core entropy zero. We give several characterizations of polynomials with core entropy zero. In particular, we show that a degree d post-critically finite polynomial f has core entropy zero if and only if f is in the degree d main molecule. The characterizations define several quantities which measure the complexities of polynomials with core entropy zero. We show that these measures are all comparable.

math.DS↗

arXiv2020

Levy and Thurston obstructions of finite subdivision rules

For a post-critically finite branched covering of the sphere that is a subdivision map of a finite subdivision rule, we define non-expanding spines which determine the existence of a Levy cycle in a non-exhaustive semi-decidable algorithm. Especially when a finite subdivision rule has polynomial growth of edge subdivisions, the algorithm terminates very quickly, and the existence of a Levy cycle is equivalent to the existence of a Thurston obstruction. In order to show the equivalence between Levy and Thurston obstructions, we generalize the arcs intersecting obstruction theorem by Pilgrim and Tan to a graph intersecting obstruction theorem. As a corollary, we prove that for a pair of post-critically finite polynomials, if at least one polynomial has core entropy zero, then their mating has a Levy cycle if and only if the mating has a Thurston obstruction.

math.DS↗

arXiv2020

Quasi-self-similar fractals containing "Y" have dimension larger than one

Suppose $X$ is a compact connected metric space and $f: X \to X$ is a metric coarse expanding conformal map in the sense of Ha\"issinsky-Pilgrim. We show that if $X$ contains a homeomorphic copy of the letter "Y", then the Hausdorff dimension of $X$ is greater than one. As an application, we show that for a semi-hyperbolic rational map $f$ its Julia set $\mathcal{J}_f$ is quasi-symmetric equivalent to a space having Hausdorff dimension 1 if and only if $\mathcal{J}_f$ is homeomorphic to a circle or a closed interval.

math.MG↗

arXiv2020

Haupt's theorem for strata of abelian differentials

Let S be a closed topological surface. Haupt's theorem provides necessary and sufficient conditions for a complex-valued character of the first integer homology group of S to be realized by integration against a complex-valued 1-form that is holomorphic with respect to some complex structure on S. We prove a refinement of this theorem that takes into account the divisor data of the 1-form.

math.GT↗