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Kyounghee Kim

Publications and source records attributed to Kyounghee Kim.

At least 19 recordsLinked to original sources
arXiv2025

Spectral Growth in $W(E_{10})$: Double Coset Filtration and Hilbert Geometry

We study the spectral radii of elements in the hyperbolic Coxeter group $W(E_{10})$ by introducing a filtration indexed by reflections conjugate to a distinguished simple reflection $s_0$. This filtration organizes $W(E_{10})$ into double cosets relative to the parabolic subgroup $W(A_9)$, and we classify the minimal representatives of these cosets via a rooted directed acyclic graph (DAG) labeled by triples. Each node in the DAG corresponds to a structured reflection composition, enabling a recursive understanding of spectral growth. Using the Hilbert metric on the Tits cone, we relate spectral radii to geometric displacement and demonstrate an effective method to compute the spectral radii inductively. This provides a geometric and combinatorial framework for understanding the Weyl spectrum of $W(E_{10})$. While our focus is on $E_{10}$, the techniques developed extended naturally to the family $W(E_n)$ for $n\ge 10$, with implications for dynamics on rational surfaces and entropy spectra of surface automorphisms.

math.GR↗
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↗
arXiv2024

Mapping classes of Real Rational Surface Automorphisms

Let $\{F_n, n\ge 8\}$ be a family of diffeomorphisms on real rational surfaces that are birationally equivalent to birational maps on $\mathbf{P}^2(\mathbb{R})$. In this article, we investigate the mapping classes of the diffeomorphisms $F_n, n\ge 8$. These diffeomorphisms are reducible with unique invariant irreducible curves, and we determine the mapping classes of their restrictions, $\hat F_n, n \ge 8$, on the cut surfaces, showing that they are pseudo-Anosov and do not arise from Penner's construction. For $n=8$, Lehmer's number is realized as the stretch factor of $\hat F_8$, a pseudo-Anosov map on a once-punctured genus $5$ orientable surface. The diffeomorphism $\hat F_8$ is a new geometric realization of a Lehmer's number.

math.DS↗
arXiv2023

Salem/Pisot Numbers in the Weyl Spectrum

n this article, we define orbit data for birational maps of $\mathbf{P}^2(\mathbb{C})$ and show that this data uniquely determines the dynamical degree by providing minimal polynomials for dynamical degrees in terms of orbit data. Leveraging this relationship, we recursively identify all Salem and Pisot numbers that appear in the Weyl spectrum of the union of the Coxeter groups $W_n$ associated with $E_n$ and the set of all dynamical degrees of birational maps of $\mathbf{P}^2(\mathbb{C})$. Furthermore, we demonstrate that all accumulation points of Pisot numbers less than or equal to 2 are present in the Weyl spectrum.

math.GR↗
arXiv2022

The Dynamical Degrees of Rational Surface Automorphisms

The induced action on the Picard group of a rational surface automorphism with positive entropy can be identified with an element of the Coxeter group associated to $E_n, n\ge 10$ diagram. It follows that the set of dynamical degrees of rational surface automorphisms is a subset of the spectral radii of elements in the Coxeter group. This article concerns the realizability of an element of the Coxeter group as an automorphism on a rational surface with an irreducible reduced anti-canonical curve. For any unrealizable element, we explicitly construct a realizable element with the same spectral radius. Hence, we show that the set of dynamical degrees and the set of spectral radii of the Coxeter group are, in fact, identical. This has been shown by Uehara in \cite{Uehara:2010} by explicitly constructing a rational surface automorphism. This construction depends on a decomposition of an element of the Coxeter group. Our proof is conceptual and provides a simple description of elements of the Coxeter group, which are realized by automorphisms on anti-canonical rational surfaces.

math.DS↗
arXiv2022

An Automorphism group of a rational surface: Not too big not too small

This article concerns the realization problem of subgroups of Coxeter groups. We construct a subgroup $G$ of the Coxeter group $W_{15}$ such that $G$ is realized as automorphism groups of a rational surface $X$ and $G \cong Aut(X)^* \cong D_3 \times \mathbb{Z}$. We also show that there is an element $\omega$ in $W_{14}$ is not realizable.

math.AG↗
arXiv2021

Entropy of real rational surface automorphisms: Actions on the Fundamental Groups

This article discusses a method to compute the induced action on the fundamental group of a real rational surface and provides the induced actions for basic quadratic real automorphisms. Using an invariant set in the fundamental group, we introduce a method to estimate a lower bound of the action's growth rate on the fundamental group. The growth rate of this induced action gives a lower bound for the entropy of real surface automorphism.

math.DS↗
arXiv2018

Entropy of real rational surface automorphisms

We compare real and complex dynamics for automorphisms of rational surfaces that are obtained by lifting \chg{some} quadratic birational maps of the plane. In particular, we show how to exploit the existence of an invariant cubic curve to understand how the real part of an automorphism acts on homology. We apply this understanding to give examples where the entropy of the full (complex) automorphism is the same as its real restriction. Conversely and by different methods, we exhibit different examples where the entropy is strictly decreased by restricting to the real part of the surface. Finally, we give an example of a rational surface automorphism with positive entropy whose periodic cycles are all real.

math.DS↗
arXiv2014

Pseudoautomorphisms with invariant elliptic curves

We describe an explicit method for constructing pseudo-automorphisms of a space $X$ which is obtained by blowing up points of $P^k$ (or a product $P^k \times \cdots \times P^k$). The centers of blowup are chosen to lie on an elliptic normal curve and are determined using the arithmetic on the curve. These pseudo-automorphism have dynamical degree greater than $1$.

math.DS↗
arXiv2013

Pseudo-automorphisms with no invariant foliation

We construct an example of a birational transformation of a rational threefold for which the first and second dynamical degrees coincide and are $>1$, but which does not preserve any holomorphic (singular) foliation. In particular, this provides a negative answer to a question of Guedj. On our way, we develop several techniques to study foliations which are invariant under birational transformations.

math.DS↗
arXiv2012

Dynamics of (Pseudo) Automorphisms of 3-space: Periodicity versus positive entropy

We study the iterative behavior of the family of 3-step linear fractional recurrences and the family of birational maps they define. We determine all the possible periodicities within this family or, equivalently, the birational maps of finite order. This family also contains pseudo-automorphisms of infinite order. One such family consists of completely integrable maps, and another family consists of maps of positive entropy. Both of these families have invariant families of $K3$ surfaces.

math.DS↗
arXiv2009

Linear Fractional Recurrences: Periodicities and Integrability

We consider k-step recurrences of the form $z_{n+k} = A(z)/B(z)$, where A and B are linear functions of $z_n, z_{n+1}, ..., z_{n+k-1}$, which we call k-step linear fractional recurrences. The first Theorem in this paper shows that for each k there are k-step linear fractional recurrences which are periodic of period 4k. Among this class of recurrences, there is also the so-called Lyness process, which has the form $A(z)/B(z) = (a +z_{n+1} + z_{n+2} + ... + z_{n+k-1})/z_n$. The second Theorem shows that the Lyness process has quadratic degree growth. The Lyness process is integrable, and we discuss its known integrals.

math.DS↗
arXiv2009

Dynamics of Rational Surface Automorphisms: Rotation Domains

We consider rational surface automorphisms with positive entropy. A Fatou component is said to be a rotation domain if the automorphism induces a torus action on it. Here we construct a rational surface automorphism with positive entropy with the following property: it has a rotation domain which contains both a curve of fixed points and isolated fixed points. This Fatou component cannot be imbedded into complex euclidean space, so we introduce a global linear model space and show that it can be globally linearized in this model.

math.DS↗
arXiv2009

Dynamics of Rational Surface Automorphisms: Linear Fractional Recurrences

We consider the family $f_{a,b}(x,y)=(y,(y+a)/(x+b))$ of birational maps of the plane and the parameter values $(a,b)$ for which $f_{a,b}$ gives an automorphism of a rational surface. In particular, we find values for which $f_{a,b}$ is an automorphism of positive entropy but no invariant curve. The Main Theorem: If $f_{a,b}$ is an automorphism with an invariant curve and positive entropy, then either (1) $(a,b)$ is real, and the restriction of $f$ to the real points has maximal entropy, or (2) $f_{a,b}$ has a rotation (Siegel) domain.

math.DS↗
arXiv2009

Continuous Families of Rational Surface Automorphisms with Positive Entropy

We construct k-parameter families of rational surface automorphisms for any k. These are automorphisms of surfaces X, which are constructed from iterated blowups over the projective plane. In certain cases: we are able to determine the exact automorphism group of X, as well as when two of the surfaces X are inequivalent.

math.CV↗
arXiv2007

The derivatives of Asian call option prices

The distribution of a time integral of geometric Brownian motion is not well understood. To price an Asian option and to obtain measures of its dependence on the parameters of time, strike price, and underlying market price, it is essential to have the distribution of time integral of geometric Brownian motion and it is also required to have a way to manipulate its distribution. We present integral forms for key quantities in the price of Asian option and its derivatives ({\it{delta, gamma,theta, and vega}}). For example for any $a>0$ $\mathbb{E} [ (A_t -a)^+] = t -a + a^{2} \mathbb{E} [ (a+A_t)^{-1} \exp (\frac{2M_t}{a+ A_t} - \frac{2}{a}) ]$, where $A_t = \int^t_0 \exp (B_s -s/2) ds$ and $M_t =\exp (B_t -t/2).$

q-fin.PR↗