Advanced Studies Institute in Analysis on Fractal Spaces & Dynamical Systems
Date: August 4-13, 2023
Venue: Urgench State University (Uzbekistan)
Contact: Zair Ibragimov (California State University, Fullerton)
E-mail:
zibragimov@fullerton.edu
Key Lecturer: Mario Bonk
University of California
Los Angeles
Overview
A Thurston map is a branched covering map on a topological 2-sphere for which the forward orbit of each critical point under iteration is finite. Each such map gives rise to a fractal geometry on its underlying 2-sphere. The study of these maps links diverse area of mathematics such as dynamical systems, classical conformal analysis, Teichmüller theory, and analysis on metric spaces. In my lectures I will give an introduction to this circle of ideas.
Topics
- Branched coverings and Thurston maps
- Thurston’s characterization of postcritically-finite rational maps
- Visual metrics of expanding Thurston maps
- Basic notions of quasiconformal geometry
- The visual sphere of an expanding Thurston map
Key Lecturer: Mikhail Lyubich
State University of New York
Stony Brook
Overview
Holomorphic Dynamics is a theory of iteration of rational endomorphisms of
the Riemann sphere. This theory was founded in the early 20th century in
the classical work of Fatou and Julia and has become a central theme in
Dynamics in the past fourty years. Methods and ideas of the classical
geometric function theory, hyperbolic geometry, and ergodic theory have
found a beautiful interplay in this area.
Topics
- Classical Fatou-Julia Dichotomy
- Sullivan's No Wandering Domains Theorem
- Structural Stability and lambda-lemma
- Basic structure of the Mandelbrot set
- Distribution of preimages and periodic points
Invited Lecturers
Sergiy Merenkov
City College of New York and CUNY Graduate Center
Title:
Geometry and dynamics versus regularity for manifold automorphisms.
Abstract: I plan to discuss how regularity of automorphisms of manifolds influences the geometry and dynamics of their minimal sets and vice versa. A particular attention will be given to smooth and quasiconformal homeomorphisms of n-tori, n>=2, and hyperbolic surfaces. The results to be presented are higher dimensional versions of Denjoy's theorem on topological conjugacy for circle diffeomorphisms.
Pekka Pankka
University of Helsinki, Finland
Title:
From quasiregular mappings to quasiregular curves.
Abstract: Quasiregular mappings are a non-injective quasiconformal mappings and they may be viewed as a higher dimensional generalization of holomorphic mappings. I will first discuss some of the recent results in their global theory. Then I will discuss a new class of mappings generalizing both quasiregular mappings and holomorphic curves, called quasiregular curves.
Dzmitry Dudko
State University of New York at Stony Brook
Title:
From Denjoy's Theorem to the Renormalization Theory.
Abstract:
Renormalization is a powerful tool for understanding dynamical systems on various scales. We will introduce some of the renormalization ideas, often illustrating them on pictures. In the first talk, we will review Denjoy's Theorem on rotation number and then proceed to finer renormalization questions involving circle maps. In the second talk, we will discuss the period-doubling renormalization of real quadratic polynomials and make an analogy between the associated adding machine attractor and circle maps.
Sabya Mukherjee
Tata Institute of Fundamental Research, India
Title: Fractals arising from antiholomorphic maps and reflection groups.
Abstract: We will explicate topological and geometric structures of fractal sets arising from iterations of antiholomorphic rational maps (complex conjugates of rational maps of the Riemann sphere) and Kleinian reflection groups. We will also explain, mostly at the level of examples, how these conformal dynamical systems can be combined to manufacture richer systems with unusual dynamical and complex-analytic properties.
Zhiqiang Li
Peking University, China
Title: Introduction to ergodic optimization and its connection to fractal spaces
Abstract: Ergodic optimization is a young area in ergodic theory, closely related to thermodynamic formalism. It studies a special class of measures, called maximizing measures, and has close connections to various other parts of mathematics. In this mini-course, we will give an introduction to ergodic optimization and discuss some recent developments in connection to fractal spaces arising from complex dynamics.
Program Schedule
August 4, 2023
11:00 – 11:30 Registration
11:30 – 12:00 Opening Remarks
Zair Ibragimov - California State University, Fullerton
Bakhrom Abdullaev - Rector, Urgench State University
Misha Lyubich - State University of New York, Stony Brook
Mario Bonk - University of California, Los Angeles
12:00 – 13:00 Lunch
13:00 – 14:30 Rest (Faculty Housing)
14:30 – 15:20 Speaker: Mario Bonk
Title: Branched coverings and Thurston maps, I
15:30 – 16:20 Speaker: Mario Bonk
Title: Branched coverings and Thurston maps, I
16:30 – 17:00 Coffee Break
17:00 – 17:50 Speaker: Sergiy Merenkov
Title: Geometry and dynamics vs. regularity for manifold automorphisms, I
18:00 – 18:50 Speaker: Sergiy Merenkov
Title: Geometry and dynamics vs. regularity for manifold automorphisms, II
19:30 – 21:30 Dinner
August 5, 2023
10:00 – 10:50 Speaker: Mario Bonk
Title: Thurston’s characterization of postcritically-finite rational maps
11:00 – 11:50 Speaker: Mario Bonk
Title: Visual metrics of expanding Thurston maps
12:00 – 12:30 Coffee Break
12:30 – 13:20 Speaker: Zhiqiang Li
Title: Introduction to ergodic optimization and connection to fractal spaces, I
13:30 – 14:30 Lunch
14:30 – 17:00 Rest (Faculty Housing)
17:00 – 17:50 Speaker: Pekka Pankka
Title: From quasiregular mappings to quasiregular curves, I
18:00 – 18:50 Speaker: Pekka Pankka
Title: From quasiregular mappings to quasiregular curves, II
19:30 – 21:30 Dinner
August 6, 2023 (Khorazm Mamun Academy)
08:00 – 08:50 Speaker: Mario Bonk
Title: Basic notions of quasiconformal geometry
09:00 – 09:50 Speaker: Mario Bonk
Title: The visual sphere of an expanding Thurston map
10:00 – 10:30 Tour of Khorezm Mamun Academy
10:30 – 11:20 Speaker: Zhiqiang Li
Title: Introduction to ergodic optimization and connection to fractal spaces II
12:00 – 14:00 Lunch (Ichan Kala)
15:00 – 19:00 Guided Tour of Ichan Kala
19:30 – 22:00 Banquet for Mario (Ichan Kala)
August 7, 2023
10:00 – 10:50 Speaker: Misha Lyubich
Title: Classical Fatou-Julia Dichotomy, I
11:00 – 11:50 Speaker: Misha Lyubich
Title: Classical Fatou-Julia Dichotomy, II
12:00 – 12:30 Coffee Break
12:30 – 13:30 Recitation Session (Moderator: Zhiqiang Li)
13:30 – 14:30 Lunch
14:30 – 17:00 Rest (Faculty Housing)
17:00 – 17:50 Speaker: Dzmitry Dudko
Title: From Denjoy’s Theorem to the Renormalization Theory, I
18:00 – 18:50 Speaker: Dzmitry Dudko
Title: From Denjoy’s Theorem to the Renormalization Theory curves, II
19:30 – 21:30 Dinner
August 8, 2023
10:00 – 10:50 Speaker: Misha Lyubich
Title: Classical Fatou-Julia Dichotomy, III
11:00 – 11:50 Speaker: Misha Lyubich
Title: Structural Stability and lambda-lemma
12:00 – 12:30 Coffee Break
12:30 – 13:30 Recitation Session (Moderators: Zhiqiang Li and Timothy Alland)
13:30 – 14:30 Lunch
14:30 – 17:00 Rest (Faculty Housing)
17:00 – 17:50 Speaker: Sabyasachi Mukherjee
Title: Fractals arizing from antiholomorphic maps and reflection groups, I
18:00 – 18:50 Speaker: Sabyasachi Mukherjee
Title: Fractals arizing from antiholomorphic maps and reflection groups, II
19:30 – 21:30 Dinner
August 9, 2023
08:00 – 13:00 Trip to Ayaz Kala
13:30 – 14:30 Lunch
14:30 – 16:00 Rest (Faculty Housing)
16:00 – 17:00 Recitation Session (Moderator: Timothy Alland)
17:00 – 17:50 Speaker: Misha Lyubich
Title: Basic structure of the Mandelbrot set
18:00 – 18:50 Speaker: Misha Lyubich
Title: Distribution of preimages and periodic points
19:30 – 22:00 Banquet for Misha (Urgench)
August 10-13, 2023
Trips to Bukhara (August 10-11), Samarkand (August 11-12) and Tashkent (August 12-13)
List of Student Participants
- Alberto Acevedo (The University of Arizona)
- Nurali Akramov (National University of Uzbekistan)
- Klara Allaberganova (Novosibirsk State University)
- Timothy Alland (State University of New York at Stony Brook)
- Yuke An (Peking University)
- Reginald Anderson (Kansas State University)
- Jinru Bai (Peking University)
- Sardor Bekchanov (Urgench State University)
- Katherine Booth (Georgia Institute of Technology)
- Rasiel Chishti (Pennsylvania State University)
- Carmen Day (California State University, Channel Islands)
- Abigail D'Ovidio Long (University of Nebraska-Lincoln)
- Kutlimurat Erkinboyev (Urgench State University)
- Davis Funk (West Virginia University)
- Nicholas Hanson (University of Illinois, Chicago)
- Qiangdu He (Peking University)
- Yinying Huang (Peking University)
- Okhunjon Ismoilov (Romanovski Institute of Mathematics)
- Mukhiddin Ismoilov (National University of Uzbekistan)
- Ashley Jubb (California State University, Fullerton)
- Maxliyo Kadirova (Novosibirsk State University)
- Xursandbek Kamolov (Urgench State University)
- Xiaoran Li (Peking University)
- Mohammed Mannan (New York University)
- Alan Mitchell (University of California, Irvine)
- Aditya Mohan (University of California, Santa Barbara)
- Eden Nava-Martinez (California State University, Long Beach)
- Salvador Ochoa Zavalza (University of California, Davis)
- Karen Padgett (California State University, Fullerton)
- Ruicen Qiu (Peking University)
- Tanner Reese (The University of Arizona)
- Cynthia Rivera Sánchez (University of Puerto Rico)
- Rafael Saavedra (Harvard University)
- Xianghui Shi (Peking University)
- G‘ulomjon Sotliqov (Urgench State University)
- Yiqing Sun (Peking University)
- William Wood (University of California, Irvine)
- Ranfeng Yu (Peking University)
- Hanyun Zheng (Peking University)
- Qu Zhengbo (Peking University)
