Zhiqiang Li
Beijing, China
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Office
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BICMR 78105W-1
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Office Hours
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By appointment
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E-Mail
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zli at math .pku .edu .cn
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Positions
I got my
PhD in 2015 from UCLA under the supervision of Professor Mario Bonk.
Teaching
Seminars
Pacific Quasiworld
(organized by Mario Bonk, Yunping Jiang, Zhiqiang Li, and Mitsuhiro Shishikura)
Conformal dynamics and groups (PKU, organized
by Zhiqiang Li, Junyi Xie, and Wenyuan Yang)
Undergraduate 3+X Dynamics Seminar - Complex
Analysis, Geometry, and Dynamics (Fall 2022 - current) (PKU, organized by Zhiqiang Li)
Prof. Wen s seminar (PKU)
Past
Seminars
Undergraduate
Dynamics Seminar (Spring 2022) (PKU,
organized by Wenyuan Yang and Zhiqiang Li)
Undergraduate
Dynamics Seminar (Fall 2020 - Spring 2021) (PKU, organized by Shaobo Gan, Disheng Xu, and Zhiqiang Li)
May 17 - June 11 2023 Kasra
Rafi (University of Toronto)
June 3-13 2023 Mario Bonk (UCLA)
July 14-15 2023 Dylan Thurston
(Indiana University)
August 23 - September 7 2023 Tianyi
Zheng (UCSD)
September 6-11 2023 Oliver Jenkinson
(Queen Mary, University of London)
October 9-18 2023 Ilia
Binder (University of Toronto)
January 6-14 2024 Pekka
Pankka (University of Helsinki)
March 25-29 2024 Yunping
Jiang (City University of New York)
April 21 - May 4 2024 Daniel
Meyer (University of Liverpool)
Research
Interests
My research
interests lie in the interplay of dynamical systems and geometry of spaces with
fractal nature.
Publications
and Preprints
Z. Li
(with Ilia Binder, Qiandu
He, Xianghui Shi), On computability of
equilibrium states beyond uniform expansion. In preparation.
Z. Li (with Juan Rivera-Letelier), Prime orbit theorems for Collet–Eckmann maps. In
preparation.
Z. Li (with Pekka
Pankka and Hanyun Zheng) Bonk–Meyer
theorem for postcritically-finite uniformly quasiregular maps. In preparation.
Z. Li (with Zelai Hao, Yinying Huang),
Ergodic optimization for the β-transformations. In preparation.
Z. Li (with Yinying Huang), Ground states and periodic orbits for the Gauss map. In
preparation.
Z. Li (with Ruicen Qiu),
General harmonic measures for distance-expanding dynamical systems. Available upon request.
Z. Li (with Mario Bonk and Wenbo Li), On dynamical Kapovich–Kleiner conjecture. Available
upon request.
Z. Li (with Xianghui Shi), Entropy approachability and large deviation
principles without upper-semicontinuity of entropy. Available upon request.
Z. Li (with Xianghui Shi), Thermodynamic formalism for subsystems of expanding
Thurston maps II. Submitted, (arXiv:2404.07247). 46 pages. (PDF)
Z. Li (with Xianghui
Shi, Yiwei Zhang), Thermodynamic formalism for subsystems of expanding
Thurston maps and large deviations asymptotics. Submitted, (arXiv: 2312.15822). 57 pages. (PDF)
Z. Li (with Xianghui Shi), A local central limit theorem for expanding Thurston
maps. Available upon request.
Z. Li (with Xiaoran Li, Yiwei Zhang), Thermodynamic formalism for correspondences. Submitted, (arXiv: 2311.09397). 100 pages. (PDF)
Z. Li (with Hanyun Zheng), Weak expansion properties and a large deviation
principle for coarse expanding conformal systems. Submitted, (arXiv:2311.07305). 38 pages. (PDF)
Z. Li (with Jian Ding, Yiwei Zhang), On the prevalence of the periodicity of maximizing
measures. Adv. Math. 438, 2024, 109485. 23 pages. (PDF)
Z. Li (with Yiwei Zhang), Ground states and periodic orbits for expanding
Thurston maps. Submitted, (arXiv:2303.00514). 51
pages. (PDF)
Z. Li (with Tianyi Zheng), Prime orbit theorems for expanding Thurston maps: Genericity of strong
non-integrability condition. Submitted, (arXiv: 2312.06687). 27 pages. (PDF)
Z. Li (with Tianyi
Zheng), Prime orbit theorems for expanding Thurston maps: Lattès maps and split
Ruelle operators. Submitted, (arXiv: 2312.06688). 62 pages. (PDF)
Z. Li (with Tianyi
Zheng), Prime orbit theorems for expanding Thurston maps: Dirichlet series and
orbifolds. Adv. Math. 443, 2024, 109600.
89 pages. (PDF)
Z. Li
(with Tianyi Zheng), Prime orbit theorems
for expanding Thurston maps. Not intended for publication, (arXiv:1804.08221). 168 pages
complete version (PDF), 79
pages abridged version (PDF)
Z. Li, Ergodic theory of expanding Thurston maps,
volume 4 of The Atlantis Series in Dynamical
Systems, Atlantis Press (Springer), 2017. 191 pages. (PDF)
Z. Li,
Weak expansion properties and large deviation principles for expanding Thurston
Maps. Adv. Math. 285 (2015), 515
–567. (PDF)
Z. Li,
Equilibrium states for expanding Thurston maps. Comm. Math. Phys.
357 (2018), 811–872. (PDF)
Z. Li,
Periodic points and the measure of maximal entropy of an expanding Thurston
map. Trans.
Amer. Math. Soc., 368 (2016), 8955–8999. (PDF)
Z. Li
(with Balasingham
Balamohan and Stephen
Tanny), A Combinatorial Interpretation for Certain Relatives
of the Conolly Sequence. J. Integer Seq., 11 (2008), Article
08.2.1. (PDF).
Student
Research
Graduate Students
Hongfei Liu graduated with a Master's degree in 2023 and
worked with me on local central limit theorems for topological Collet–Eckmann maps.
Yinying Huang has been working with me on the Typically
Periodic Optimization Conjecture in a countable setting. A manuscript is in
preparation. Yinying has also been working with Zelai Hao and me on the
Typically Periodic Optimization Conjecture for the β-transformations. He is also interested in topological
Collet-Eckmann maps and random dynamical systems.
Xianghui Shi has been working with me and Yiwei Zhang on
thermodynamical formalism for subsystems of expanding Thurston maps, inspiring
a translation of the notion of subgroups from geometric group theory via
Sullivan's dictionary. As an application, fine large deviation properties for
expanding Thurston maps were obtained. See the preprint here.
Xianghui has been discussing with Mario Bonk and me on formalism for the
conformal dimension, among other topics of his interests.
Ranfeng Yu has been working on the intersection between
metric geometry, probability, and dynamics. He is currently working with Tianyi
Zheng and me on quasi-isometries of percolations. Ranfeng is also talking with
Mario Bonk on this topic. On the other hand, Ranfeng is thinking about the
prevalence of periodic optimization following my work with Jian Ding and Yiwei
Zhang.
Undergraduate Students
Qiandu He has been working with me, Ilia Binder, and
Yiwei Zhang on the computability in thermodynamical formalism. A manuscript in
the uniformly expanding case, including applications to hyperbolic rational
maps, is available here.
Qiandu is now thinking about new projects with Ilia Binder and me. On the other
hand, Qiandu is starting a new project on the computability of equilibrium
states in SCV under the supervision of Fabrizio Bianchi, Michael Yampolsky, and
me. Qiandu is also thinking about the computability of maximizing measures with
Yinying Huang and me.
Jie Ji has been studying invariant graphs on the
plane under the supervision of J. Rivera-Letelier and me. See his slides here.
Xiaoran Li has been working with Yiwei Zhang and me to
extend thermodynamic formalism to correspondences and Markov processes. A
preprint is available here.
Xiaoran has been working with a graduate student, Nils Hemmingsson, from
Sweden, on a project on the conformal measure for correspondences under the
supervision of Misha Lyubich and I. Xiaoran is also exploring another new
project under the supervision of Sabyasachi Mukherjee and me.
Ruicen Qiu has been working with Manfred Denker, me,
and Yiwei Zhang on the study of the Martin boundary on the dynamic side of
Sullivan's dictionary. A manuscript in the uniformly expanding case is ready.
The study in the setting of expanding Thurston maps is ongoing. Ruicen has also
been thinking about the Finite Curve Attractor Conjecture with Mario Bonk and
me.
Hanyun Zheng has been working with me on weak expansion
properties and their applications in thermodynamical formalism for coarse
expanding conformal systems. See his slides here.
A preprint is available here.
Hanyun has been working with Pekka Pankka and me on building a theory of PCF
uniformly quasiregular maps. Hanyun is still thinking about CXC systems.
Yiqing Sun has been working on ergodic optimization.
Jinru Bai has been working on geometry and Sullivan's
dictionary with Mario Bonk and me, focusing on the solenoid.
Yuke An has been working on geometry
and Sullivan's dictionary with Mario Bonk and me, focusing on the Finite
Curve Attractor Conjecture.
Zhengbo Qu has been working on the amenability of the
iterated monodromy groups for expanding Thurston maps with Tianyi Zheng and me.
Zelai Hao has been working on ergodic optimization for
the β-transformations with Yinying Huang and me. A preprint is close
to completion.
Available
projects for undergraduate students
(1)
Correspondences
and their applications in higher dimensions, with potential involvement of
Misha Lyubich.
(2)
Correspondences
and their applications in complex billiards, with potential involvement of
Misha Lyubich.
Prospective postdocs,
graduate students, and undergraduate students
There are opportunities for postdoctoral
positions, graduate study supervision, and undergraduate research projects.
Please get in touch with me via email or in person.
About Me
Miscellaneous
A Poem in Chinese (PDF)