Location: SIMIS R610, SIMIS
Zoom Meeting ID: 844 0594 7424 (Passcode: 076895)
Schedule: Wednsday, Sep. 10, 2025
- 10:30 – 11:20 – Zin Arai
- 11:30 – 12:20 – Valerii Sopin
- 12:30 – 13:20 – Asad Ullah
- Lunch break
- 14:30 – 15:20 – Eran Igra
- 15:30 – 16:20 – Yanghong Yu
- 16:30 – 17:20 – Bowen Chen
- 17:30 – 18:20 – Michał Kowalewski
Abstracts:
Title: On the topology of the higher-dimensional Mandelbrot set, Zin Arai (Institute of Science Tokyo)
We study the parameter space of the real and complex Hénon map from a topological point of view. As in the case of the one-dimensional quadratic map, the bifurcation of the Hénon map is governed by the higher-dimensional Mandelbrot set. However, in this 2-dimensional setting, the topology of the Mandelbrot set becomes much more complicated than the original one-dimensional case.
We develop a monodromy theory associated with the Mandelbrot set and describe its topological complexity in terms of the monodromy representation.
Title: Moduli space in 2-Morse Theory, Bowen Chen (SIMIS)
2-Morse theory is the study of 2 commuting Morse vector fields on Riemannian manifold. It can be seen as a family of gradient equations which is a hybrid of Morse type system and Morse-Bott type system. We define the moduli space of film which generalizes the moduli space of gradient lines in Morse theory. We show that each solution to the family of gradient equations is somehow a convex polygon. In this talk, we will review some dynamics of gradient flow in standard Morse theory, then investigate the dynamics of the film in 2-Morse theory. We end up with the dimension counting fomula for general 2-Morse moduli space and point out the relation between 2-Morse theory with convex polygonal subdivision.
Title: Kneading the Lorenz attractor, Eran Igra (SIMIS)
One (useful) way to study three dimensional chaotic attractors is to define interval maps which behave similarly (and are often and are easier to analyze qualitatively), and then compare their dynamics and bifurcations with the numerical observations of the Differential Equation. Of course, chaotic attractors are not suspended one-dimensional maps, which raises an interesting question – can we explain analytically why their dynamics and bifurcations are described so well by one-dimensional maps? In this talk we will answer this question for the Lorenz attractor. As we will prove, one can rigorously reduce the butterfly attractor into a one-dimensional interval map which captures all its essential dynamics. Following that, using renormalization theory we will prove how one can use these results to give “topological lower bounds” for the possible complexity of knots realized as periodic orbits on the attractor. Based on joint work with Łukasz Cholewa.
Title: Tranched graphs: consequences for topology and dynamics, Michał Kowalewski (AGH University)
The talk will introduce the main results from the preprint under the same name (arXiv:2405.05407), coauthored with Piotr Oprocha. We will introduce the definitions of objects that motivated the study of tranched graphs, all of which are generalizations of topological graphs, containing the Warsaw circle as a non-trivial element. After that, we discuss how complex can the structure of these spaces be, as well as how it restricts admissible dynamics.
Title: Using Erdős’s methods to study Yorke’s problems, Valerii Sopin (SIMIS)
In this talk, we study the possible bifurcations of periodic orbits by reducing them to graphs. The aforementioned allows to study the genericity of routes to chaos, as well as to analyze their possible complexity by applying graph-theoretic tools. In particular, we will prove there is no upper bound on the possible complexity of routes to chaos, which correlates well with several numerical studies. Time permitting, we will also discuss how our results suggest general fermionic description of bifurcations via the virial expansion, and a universal description via the Rado graph.
Title: Statistical properties of dynamical systems via induced Weak Gibbs Markov maps, Asad Ullah (SIMIS)
In this talk, I will discuss the existence of a mixing invariant probability measure, absolutely continuous with respect to a reference measure, for dynamical systems that admit induced Weak Gibbs Markov maps. I will also present results on the decay of correlations, the Central Limit Theorem, and Large Deviations for such dynamical systems. The main results presented in this talk are detailed in [1, 2].
[1] Ullah, A., Vilarinho, H. Statistical properties of dynamical systems via induced weak Gibbs Markov maps. Nonlinearity 38 (2025), 045024.
[2] Ullah, A., Vilarinho, H. Decay of Correlations via induced Weak Gibbs Markov maps for nonHolder observables. arXiv:2403.09528v2 (2024).
Title: Transition matrix without continuation in the Conley index theory, Yanghong Yu (Institute of Science Tokyo)
Given a one-parameter family of flows over a parameter interval Λ, assuming there is a continuation of Morse decompositions over Λ, Reineck (1988) defined a singular transition matrix to show the existence of a connecting orbit between some Morse sets at some parameter points in Λ. This talk aims to extend the definition of a singular transition matrix in cases where there is no continuation of Morse decompositions over the parameter interval. This extension will help study the bifurcation associated with the change of Morse decomposition from a topological dynamics viewpoint.
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