Papers and preprints
The arXiv version may be slightly outdated.
19. Joint with F. Bianchi, A Ruelle-McMullen formula for the volume dimension of skew products in C^2. [arXiv:2603.06475].
Abstract: Ruelle gave an explicit second-order expansion at c=0 of the Hausdorff dimension of the Julia set of the quadratic family f_c(z)=z^2+c. McMullen later extended this result to polynomial perturbations of z^d for arbitrary degree d≥2. In this paper we study an analogue of this problem for skew products in C^2. Since holomorphic dynamical systems in higher dimensions are non-conformal, we replace the Hausdorff dimension by the \emph{volume dimension}, a dynamically defined notion we introduced in our earlier work and characterized as the zero of a natural pressure function. We consider families of holomorphic skew products of the form f_t(z,w)=(z^d,w^d+t(c_1(z)w^{d−1}+c_2(z)w^{d−2}+⋯+c_d(z))). Our main result gives an explicit second-order expansion of the volume dimension of the Julia set J(f_t) as t→0 in terms of the coefficients c_k(z).
18. Joint with N. Dalaklis, Multifractal analysis of equilibrium states of endomorphisms of P^k. [arXiv:2512.08696]
Abstract: Let f be a holomorphic endomorphism of CP^k of algebraic degree at least 2 and let X⊆CP^k be an uniformly expanding set. In this paper, we study multifractal analysis of equilibrium states of Hölder continuous functions for the non-conformal dynamical system f:X→X. In lieu of Hausdorff dimensions, we use a new dimension theory (i.e., the volume dimension theory) to define various local dimension multifractal spectra and show that each of these spectra form a Legendre transform pair with the temperature function as in the conformal case. As an application of our main theorems, we also prove a conditional variational principle for such dimension multifractal spectra.
17. Joint with F. Bianchi, Analyticity of the Hausdorff dimension and metric structures on Misiurewicz families of polynomials. [arXiv:2508.12493]
Abstract: Consider a holomorphic family (f_λ)_{λ∈Λ} of polynomial maps on C with the property that a critical point of f_λ is persistently preperiodic to a repelling periodic point of f_λ. Let Ω be a bounded stable component of Λ with the property that, for all λ∈Ω, all the other critical points of f_λ belong to attracting basins. In this paper, we introduce a dynamically meaningful geometry on Ω by constructing a natural path metric on Ω coming from a 2-form ⟨⋅,⋅⟩_G. Our construction uses thermodynamic formalism. A key ingredient is the spectral gap of adapted transfer operators on suitable Banach spaces, which also implies the analyticity of ⟨⋅,⋅⟩_G on the unit tangent bundle of Ω. As part of our construction, we recover a result of Skorulski and Urbański stating that the Hausdorff dimension of the Julia set of f_λ varies analytically over Ω.
16. Joint with H. Lee and I. Park, Complex analytic theory on the space of Blaschke products: Simultaneous
uniformization and the pressure metric. [arXiv:2507.17077]
Abstract: In this paper, we study complex analytic aspects of the moduli space Bdfm of degree d≥2 fixed-point-marked Blaschke products. We define a complex structure on B_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 SA_d^{fm}, which is a codimension-1 subspace of B_d^{fm}.
15. Joint with C. Wu, Relative train tracks and generalized endperiodic graph maps. [arXiv:2408.13401]
Abstract: We study endperiodic maps of an infinite graph with finitely many ends. We prove that any such map is homotopic to an endperiodic relative train track map. Moreover, we show that the (largest) Perron-Frobenius eigenvalue of the transition matrix is a canonical quantity associated to the map.
14. Joint with H. Nie, On a metric view of the polynomial shift locus. [arXiv: 2311.11399]
Abstract: We relate generic points in the shift locus S_D of degree D≥2 polynomials to metric graphs. Using thermodynamic metrics on the space of metric graphs, we obtain a distance function ρ_D on S_D. We study the (in)completeness of the metric space (S_D,ρ_D). We prove that when D≥3, the space (S_D,ρ_D) is incomplete and its metric completion contains a subset homeomorphic to the space PSTD∗ introduced by DeMarco and Pilgrim. This provides a new way to understand the space PSTD∗.
13. Joint with C. Wu, Basmajian-type identities over non-Archimedean local fields. To appear Trans. Amer. Math. Soc. [arXiv:2212.09992]
Abstract: Let Σ be a connected compact oriented surface with boundary and negative Euler characteristic. Let k be a non-Archimedean local field. In this paper, we prove Basmajian's identity for projective Anosov representations ρ :π1Σ→PSL(d,k), d≥2. Our series identity exhibits a drastic difference from all the Basmajian-type identities over the Archimedean fields R and C. In particular, the series is a signed finite sum. When d=2, we give a geometric proof of the identity using Berkovich hyperbolic geometry.
12. Joint with F. Bianchi, Manhattan curves in complex dynamics and asymptotic correlation of multiplier spectra. To appear Contemp. Math. [arXiv:2508.12490]
Abstract: The Manhattan curve for a pair of hyperbolic structures (possibly with cusps) on a given surface is a geometric object that encodes the growth rate of lengths of closed geodesics with respect to the two different hyperbolic metrics. It has been extensively studied as a way to understand geodesics on surfaces, the thermodynamic formalism of the geodesic flows and comparison of hyperbolic metrics. Via Sullivan's dictionary, in this paper, we define and study the Manhattan curve for a pair of hyperbolic rational maps on CP^1, and more generally of holomorphic endomorphisms of CP^k. We discuss several counting results for the multiplier spectrum and show that the Manhattan curve for two holomorphic endomorphisms is related to the correlation number of their multiplier spectra.
11. Joint with F. Bianchi, Pressure path metrics for parabolic families of polynomials. To appear Indiana Univ. Math. J. [arXiv:2409.10462]
Abstract: Let Λ be a subfamily of the moduli space of degree D≥2 polynomials defined by a finite number of parabolic relations. Let Ω be a bounded stable component of Λ with the property that all critical points are attracted by either the persistent parabolic cycles or by attracting cycles in C. We construct a positive semi-definite pressure form on Ω and show that it defines a path metric on Ω. This provides a counterpart in complex dynamics of the pressure metric on cusped Hitchin components recently studied by Kao and Bray-Canary-Kao-Martone.
10. Joint with H. Lee and I. Park, Pressure metrics in geometry and dynamics. To appear Shishikura Proceedings. [arXiv:2407.18441]
Abstract: 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.
9. Joint with F. Bianchi, A Mañé-Manning formula for expanding measures for endomorphisms of P^k. Trans. Amer. Math. Soc. 377 (2024), no. 11, 8179–8219. [arXiv:2308.03013]
Abstract: Let k≥1 be an integer and f a holomorphic endomorphism of Pk(C) of algebraic degree d≥2. We introduce a volume dimension for ergodic f-invariant probability measures with strictly positive Lyapunov exponents. In particular, this class of measures includes all ergodic measures whose measure-theoretic entropy is strictly larger than (k−1)logd, a natural generalization of the class of measures of positive measure-theoretic entropy in dimension 1. The volume dimension is equivalent to the Hausdorff dimension when k=1, but depends on the dynamics of f to incorporate the possible failure of Koebe's theorem and the non-conformality of holomorphic endomorphisms for k≥2. If \nu is an ergodic f-invariant probability measure with strictly positive Lyapunov exponents, we prove a generalization of the Mañé-Manning formula relating the volume dimension, the measure-theoretic entropy, and the sum of the Lyapunov exponents of \nu. As a consequence, we give a characterization of the first zero of a natural pressure function for such expanding measures in terms of their volume dimensions. For hyperbolic maps, such zero also coincides with the volume dimension of the Julia set, and with the exponent of a natural (volume-)conformal measure. This generalizes results by Denker-Urbański and McMullen in dimension 1 to any dimension k≥1. Our methods mainly rely on a theorem by Berteloot-Dupont-Molino, which gives a precise control on the distortion of inverse branches of endomorphisms along generic inverse orbits with respect to measures with strictly positive Lyapunov exponents.
8. Joint with L. Chen, Non-realizability of some big mapping class groups. Proc. AMS 152 (2024), 4503-4514. [arXiv:2111.08583]
Abstract: In this note, we prove that the compactly supported mapping class group of a surface containing a genus 3 subsurface has no realization as a subgroup of the homeomorphism group. We also prove that for certain surfaces with order 6 symmetries, their mapping class groups have no realization as a subgroup of the homeomorphism group. Examples of such surfaces include the plane minus a Cantor set and the sphere minus a Cantor set.
7. Joint with H. Nie, Quantitative equidistribution of angles of multipliers. Fund. Math. 257 (2022), no. 1,95–113. [pdf]
Abstract: We study angles of multipliers of repelling cycles for hyperbolic rational maps in C(z). For a fixed K≫1, we show that almost all intervals of length 2π/K in (−π,π] contain a multiplier angle with the property that the norm of the multiplier is bounded above by a polynomial in K.
6. Joint with C. Wolf, Entropy spectrum of rotation classes. J. Math. Anal. Appl. 508 (2022), no. 1, 12 pp. [pdf]
Abstract: In this note we study the entropy spectrum of rotation classes for collections of finitely many continuous potentials φ1,…,φm:X→R with respect to the set of invariant measures of an underlying dynamical system f:X→X. We show for large classes of dynamical systems and potentials that these entropy spectra are maximal in the sense that every value between zero and the maximum is attained. We also provide criteria that imply the maximality of the ergodic entropy spectra. For mmm being large, our results can be interpreted as a complimentary result to the classical Riesz representation theorem in the dynamical context.
5. Joint with H. Nie, A Riemannian metric on hyperbolic components. Math. Res. Lett. 30 (2023), no. 3, 733–764. [pdf]
Abstract: We introduce a Riemannian metric on certain hyperbolic components in the moduli space of degree d≥2 polynomials. Our metric is constructed by considering the measure-theoretic entropy of a polynomial with respect to some equilibrium state. As applications, we show that the Hausdorff dimension function has no local maximum on such hyperbolic components. We also give a sufficient condition for a point not being a critical point of the Hausdorff dimension function.
4. Identities for hyperconvex Anosov representations. J. Topol. Anal. 16 (2024), no. 4, 617-640. [pdf]
Abstract: In this paper, we establish Basmajian's identity for (1,1,2)-hyperconvex Anosov representations from a free group into PGL(n,R). We then study our series identities on holomorphic families of Cantor non-conformal repellers associated to complex (1,1,2)-hyperconvex Anosov representations. We show that the series is absolutely summable if and only if the Hausdorff dimension of the Cantor set is strictly less than one. Throughout the domain of convergence, these identities can be analytically continued and they exhibit nontrivial monodromy.
3. Prime number theorems for Basmajian-type identities. [pdf | arXiv: 1811.05367]
Abstract: We obtain asymptotic counting results with error terms for complex orthospectrum for Schottky groups and orbit counting function for quadratic polynomials. Moreover, we prove equidistribution of holonomy associated to these dynamical systems. Our results are obtained by considering generalized L-functions coming from the Basmajian-type identities introduced by the author in \cite{He}. We study the associated summatory functions using tools from analytic number theory and Thermodynamic Formalism, namely the Perron's formula and a Dolgopyat-type estimate on the spectrum of transfer operators.
2. On the displacement of generators of free Fuchsian groups. Geometriae Dedicata, 200 (2019), 255-264. [pdf]
Abstract: We prove an inequality that must be satisfied by displacement of generators of free Fuchsian groups, which is the two-dimensional version of the log(2k−1) Theorem for Kleinian groups due to Anderson-Canary-Culler-Shalen. As applications, we obtain quantitative results on the geometry of hyperbolic surfaces such as the two-dimensional Margulis constant and lengths of closed curves, which improves a result of Buser's.
1. Basmajian-type identities and Hausdorff dimension of limit sets. Ergodic Theory and Dynamical Systems, 38 (2018), 2224-2244. [pdf]
Abstract: In this paper, we study Basmajian-type series identities on holomorphic families of Cantor sets associated to one-dimensional complex dynamical systems. We show that the series is absolutely summable if and only if the Hausdorff dimension of the Cantor set is strictly less than one. Throughout the domain of convergence, these identities can be analytically continued and they exhibit nontrivial monodromy.
Not for publication
I passed my Topic Exam in March 2014 and here is my topic proposal.