Random walks and contracting elements VI: Random 3-manifolds (In preparation)
This is the sixth and the last in a series of papers concerning random walks and contracting elements. We consider random walks on the mapping class group and establish the CLT for Heegaard distance. This is building upon Maher's previous works. We also establish exponential bounds from below for the volume of a random Heegaard splitting/mapping torus, by carefully investigating Viaggi's method.
Random walks and contracting elements IV: Sublinearly Morse boundary (In preparation)
This is the fourth in a series of papers concerning random walks and contracting elements. Here we consider sublinearly Morse boundaries of a space with contracting isometries, which was defined and studied by Qing, Rafi and Tiozzo. Very recently, Gekhtman, Qing and Rafi proved that random walks on Teichmüller space and CAT(0) spaces are captured by sublinearly Morse boundaries. We recover their results with an independent approach. In particular, we prove that random walks with finite $p$-th moment are captured by $o(n^{1/p})$-Morse boundary. We also discuss the decay of the harmonic measure on these boundaries.
Contracting isometries and differentiability of the escape rate.
arXiv:2403.09992 (2024)
Using the previously developed pivoting technique for random walks with contracting isometries, we study regularity of the escape rate and the asymptotic entropy of a random walk. We prove that the escape rate of random walks is differentiable under some condition, namely, when there exists a signed measure $\eta$ such that the $|\mu_{t} - \mu_{0} - t\eta|$ is $o(t)$ in $L^{1}$-norm. The differentiability of the escape rate was previously proved by Mathieu and Sisto, under an assumption that involves the Radon-Nikodym derivative $f_{t} = d\mu_{t}/d\mu$. Moreover, by combining Gouëzel's proof of the continuity of the escape rate Chawla-Frisch-Forghani-Tiozzo's sublinear growth of entropy of displacement, we establish the continuity of the asymptotic entropy.
Random walks on groups and superlinear divergent geodesics
joint with Kunal Chawla, Vivian He and Kasra Rafi, arXiv:2310.18506 (2023)
Now there is a rich theory for random walks on groups, but this theory is not quasi-isometry invariant in general. Nonetheless, a certain group-theoretic and QI-invariant property might restrict the geometry of the group and leads to a QI-invariant statement for random walks on them. Recently, Goldsborough and Sisto provided the first result in this direction, showing that a group G containing a super-divergent element that also acts as a loxodromic on a hyperbolic space falls into this scheme. Namely, not only we have a CLT on G but CLTs on H whenever H is quasi-isometric to G. In this paper, we lift the condition that G admits an isometric action on a Gromov hyperbolic space. Namely, we show that possessing a super-divergent quasigeodesic is a QI-invariant property which guarantees the CLT.
Genericity of contracting geodesics in groups
joint with Kunal Chawla and Giulio Tiozzo, arXiv:2308.01877 (2023)
This paper deals with a cute dichotomy connecting the geometry of a finitely generated group G and the dyanmics of its elements. Namely, excluding elementary cases, G is hyperbolic (not hyperbolic, resp.) if and only if its generic elements exhibit strongly contracting dynamics. This is proved based on Wenyuan Yang's theory for counting problems on groups containing contracting elements. Also discussed is the generalization of Sullivan's logarithmic law.
Random walks and contracting elements III: Outer space and outer automorphism group
arXiv:2212.12122, submitted version (2022)
This is the third in a series of papers concerning random walks and contracting elements. We answer a question of Handel and Mosher: the expansion factors of a generic outer automorphism and its inverse are different. This is proved using the bounded geodesic image property (BGIP) of fully irreducibles and the pivoting technique. Also proved are SLLN, CLT, the converse of CLT, geodesic tracking and LDP on Outer space.
Random walks and contracting elements II: Translation lengths and quasi-isometric embedding
arXiv:2212.12119, submitted version (2022)
This is the second in a series of papers concerning random walks and contracting elements. We discuss the genericity of contracting elements and the discrepancy between the displacement and the translation length of a random isometry. We then consider the qi-embedding of a random subgroup into the space, which was proven for weakly hyperbolic groups by Taylor and Tiozzo (based on the theory of Maher and Tiozzo). We also investigate the same phenomenon for counting problems. As a byproduct, we also finish the proof of the converse of CLT, which seems new for CAT(0) spaces.
Random walks and contracting elements I: Deviation inequality and limit laws
arXiv:2207.06597, submitted version (2022)
This is the first in a series of papers concerning random walks and contracting elements. The main results are deviation inequalities and their consequences: geodesic tracking, CLT, LIL. This is done by bringing Gouëzel's pivotal time construction and Baik-Choi-Kim's pivoting technique to a more general setting (that includes Teichmüller space, Outer space, CAT(0) spaces, groups with nontrivial Floyd boundary, ...), and is a slight improvement of my previous CLT paper. By doing so, we also prove SLLN with exponential bounds and LDP.
Pseudo-Anosovs are exponentially generic in mapping class groups
arXiv:2110.06678 (2021),
to appear in Geometry and Topology
It is shown that for suitable finite generating sets, loxodromics in weakly hyperbolic spaces or mapping class groups are exponentially generic; i.e. their proportion inside the radius $R$ ball in the word metric decreases exponentially. This can be regarded as a sequel of Baik-Choi-Kim (2021) and Choi (2021), and does not rely on the automatic structure of the group.
Central limit theorem and geodesic tracking on hyperbolic spaces and Teichmüller spaces
arXiv:2106.13017 (2021),
Advances in Mathematics, Volume 431 (2023), 109236. Journal
- The purpose of this paper is to integrate the notion of pivots in Gouëzel (2021) and Baik-Choi-Kim (2021). This has a number of consequences including geodesic tracking, control on the discrepancy between displacements and translation lengths, CLT and its converse, and LIL.
- Korean version: this is not intended to be a word-to-word translation. It may present more details that are not available in Korean elsewhere.
Linear growth of translation lengths of random isometries on Gromov hyperbolic spaces and Teichmüller spaces
joint with Hyungryul Baik and Dongryul M. Kim
arXiv:2103.13616 (2021),
to appear in Journal of the Institute of Mathematics of Jussieu. Journal
The main purpose of this paper is to improve the moment condition of Dahmani-Horbez's spectral theorem (aka SLLN for translation lengths). Namely, we present an SLLN under finite first moment condition and another version of SLLN (at least linear growth) without any moment condition. The key idea is pivoting from undesirable trajectories to desirable trajectories, by designing so-called "pivots".
Simple length spectra as moduli for hyperbolic surfaces and rigidity of length identities
joint with Hyungryul Baik and Dongryul M. Kim
arXiv:2012.05652 (2020)
Abstract: In this article, we revisit classical length identities enjoyed by simple closed curves on hyperbolic surfaces. We state and prove rigidity of such identities over Teichmüller spaces. Due to this rigidity, simple closed curves with few intersections are characterised on generic hyperbolic surfaces by their lengths.
As an application, we construct a meagre set Ve in the Teichmüller space of a topological surface S, possibly of infinite type. Then the isometry class of a (Nielsen-convex) hyperbolic structure on S outside V is characterised by its unmarked simple length spectrum. Namely, we show that the simple length spectra can be used as moduli for generic hyperbolic surfaces. In the case of compact surfaces, an analogous result using length spectra was obtained by Wolpert.
On the surjectivity of the Symplectic representation of the mapping class group
joint with Hyungryul Baik and Dongryul M. Kim
arXiv:2008.10142 (2020),
Topology and its Applications, Volume 322 (2022), 108334. Journal
Abstract: In this note, we study the symplectic representation of the mapping class group. In particular, we discuss the surjecivity of the representation restricted to certain mapping classes. It is well-known that the representation itself is surjective. In fact the representation is still surjective after restricting on pseudo-Anosov mapping classes. However, we show that the surjectivity does not hold when the representation is restricted on orientable pseudo-Anosovs, even after reducing its codomain to integer symplectic matrices with a bi-Perron leading eigenvalue. In order to prove the non-surjectivity, we explicitly construct an infinite family of symplectic matrices with a bi-Perron leading eigenvalue which cannot be obtained as the symplectic representation of an orientable pseudo-Anosov mapping class.
Topological entropy of pseudo-Anosov maps from a typical Thurston construction
joint with Hyungryul Baik and Dongryul M. Kim
arXiv:2006.10420 (2020),
International Mathematics Research Notices, Volume 2022, No. 24, pp. 19762-19904. Journal
Abstract: In this paper, we develop a way to extract information about a random walk associated with a typical Thurston's construction. We first observe that a typical Thurston's construction entails a free group of rank 2. We also present a proof of the spectral theorem for random walks associated with Thurston's construction that have finite second moment with respect to the Teichmüller metric. Its general case was remarked by Dahmani and Horbez. Finally, under a hypothesis not involving moment conditions, we prove that random walks eventually become pseudo-Anosov. As an application, we first discuss a random analogy of Kojima and McShane's estimation of the hyperbolic volume of a mapping torus with pseudo-Anosov monodromy. As another application, we discuss non-probabilistic estimations of stretch factors from Thurston's construction and the powers for Salem numbers to become the stretch factors of pseudo-Anosovs from Thurston's construction.
Compensation of aberration and speckle noise in quantitative phase imaging using lateral shifting and spiral phase integration.
Choi I, Lee KR, Park YK, Optics Express, 25(24) pp. 30771-30779 (2017).
https://doi.org/10.1364/OE.25.030771