Titles and abstracts of posters
Yuya Arima (Nagoya): Analyticity of cusp-winding spectrum for the geodesic flow on hyperbolic surfaces
Abstract: The cusp-winding spectrum describes the growth rate of the number of cusp windings for the geodesic flow on hyperbolic surfaces with cusps by means of Hausdorff dimension. We show that the cusp-winding spectrum is real analytic for surfaces uniformized by a generalized Schottky group.
Jason Atnip (Queensland Australia): Universal gap growth for Lyapunov exponents of perturbed matrices
Abstract: In this poster we consider an arbitrary sequence of dxd matrices with norm at most one which have been perturbed by a small additive noise at each position. We show that there is a uniform gap between the top and bottom Lyapunov exponents for the products of the perturbed matrices. Furthermore, we show that the size of this gap is independent of the original sequence of matrices and depends only on the size of perturbation. In the case that d=2 or d=3 we show that this gap appears between each of the pairs of consecutive Lyapunov exponents.
Yi-Chiuan Chen (Academia Sinica Taiwan): On a question of J.D. Meiss concerning the anti-integrability for three-dimensional quadratic diffeomorphisms
Abstract: 3D quadratic diffeomorphisms with quadratic inverse generically have five independent parameters. When some parameters approach infinity, the diffeomorphisms may exhibit the so-called anti-integrability in the traditional sense of Aubry and Abramovici. That is, the dynamics of the diffeomorphisms reduce to symbolic dynamics on finite number of symbols. However, the diffeomorphisms may reduce to quadratic correspondences when parameters approach infinity, and the traditional anti-integrable limit does not deal with this situation. Meiss asked what about an anti-integrable limit for it. I will address this question in the poster.
Yuma Fujimoto (SOKENDAI): Learning in multi-memory games triggers complex dynamics diverging from Nash equilibrium
Abstract: Repeated games consider a situation where multiple agents are motivated by their independent rewards throughout learning. In general, the dynamics of their learning become complex. Especially when their rewards compete with each other like zero-sum games, the dynamics often do not converge to their optimum, i.e., the Nash equilibrium. To tackle such complexity, many studies have understood various learning algorithms as dynamical systems and discovered qualitative insights among the algorithms. However, such studies have yet to handle multi-memory games (where agents can memorize actions they played in the past and choose their actions based on their memories), even though memorization plays a pivotal role in artificial intelligence and interpersonal relationship. This study extends two major learning algorithms in games, i.e., replicator dynamics and gradient ascent, into multi-memory games. Then, we prove their dynamics are identical. Furthermore, theoretically and experimentally, we clarify that the learning dynamics diverge from the Nash equilibrium in multi-memory zero-sum games and reach heteroclinic cycles (sojourn longer around the boundary of the strategy space), providing a fundamental advance in learning in games.
Rin Gotou (Osaka): Nonlinear algebra on algebraic correspondences on projective line
Abstract: Nonlinear algebra is a nonlinear analogue of linear algebra which treats system of nonlinear polynomial equations, mainly arise in simulating dynamical systems. We express some results of invariant-theory based nonlinear-algebraic approach for dynamical systems caused by algebraic correspondences of projective line.
Masanobu Inubushi (Tokyo Science): Characterizing data assimilation in Navier-Stokes turbulence with transverse Lyapunov exponents
Abstract: Data assimilation (DA) reconstructing small-scale turbulent structures is crucial for forecasting and understanding turbulence. This study proposes a theoretical framework for DA based on ideas from chaos synchronization, in particular, the transverse Lyapunov exponents (TLEs). The analysis with TLEs characterizes a critical length scale, below which the turbulent dynamics is synchronized to the larger-scale turbulent dynamics, indicating successful DA. An underlying link between TLEs and the maximal Lyapunov exponent suggests that the critical length scale depends on the Reynolds number. Furthermore, we discuss new directions of DA algorithms based on the proposed framework.
Kanji Inui (Keio): Understanding the limit sets generated by general iterated function systems on unbounded space
Abstract: The limit sets generated by iterated function systems ( for short, IFSs ) with finitely many contractive mappings have been studied, and the ones generated by general IFSs ( in some sense ) also have been studied gradually.
Note that most of the papers consider IFSs defined on some bounded set in some sense, which deduces that the limit sets are always uniformly bounded with respect to the base points.
In this talk, we consider general IFSs defined on ( possibly unbounded ) complete metric spaces.
Then, under the natural condition, we show the existence and uniqueness ( in some sense ) of the (natural) family of the limit sets which are not uniformly bounded with respect to the base points in general.
In addition, we discuss some basic properties of the limit sets.
Isao Ishikawa (Ehime): Bounded weighted composition operators on functional quasi-Banach spaces and stability of dynamical systems
Abstract: We investigate the boundedness of weighted composition operators defined on a quasi-Banach space continuously included in the space of smooth functions on a manifold. We prove that the boundedness of weighted composition operators strongly limits the behavior of the original map, and it provides us an effective method to investigate properties of composition operators via the theory of dynamical system. As a result, we prove that only affine maps can induce a bounded weighted composition operator with non-vanishing weight on any infinite dimensional quasi-Banach space continuously included in the space of entire functions on the complex plane. We also prove any polynomial automorphisms except affine transforms cannot induce a weighted bounded composition operator on a quasi-Banach space composed of entire functions in the two-dimensional complex affine space under conditions.
Fuyuta Komura (RIKEN): *-homomorphisms between C*-algebras associated with dynamical systems
Abstract: One can construct C*-algebras from (topological) dynamical systems. It is a natural task to characterize the property of associated C*-algebras in terms of dynamical systems. In this poster, we investigate *-homomorphisms between C*-algebras associated with dynamical systems. As a result, we prove that a *-isomorphism between C*-algebras induce a homeomorphism between orbit spaces of underlying dynamical systems. This poster is based on the speaker's preprint arXiv:2302.10405.
Ziyu Liu (Nagoya): TBA
Abstract: TBA
Dmitrii Mints (ICL UK): Hidden degeneracies and flat homoclinic tangencies for multidimensional diffeomorphisms
Abstract: Our research is aimed at studying the dynamics of smooth multidimensional maps (local diffeomorphisms) from Newhouse domains, that is, open regions in the space of maps where systems with
homoclinic tangencies are dense. We prove that maps with flat homoclinic tangencies of corank-1 (invariant manifolds forming tangency of corank-1 have a unique common tangent vector) and with arbitrarily degenerate periodic orbits are dense in the Newhouse domains in the space of smooth k-dimensional maps, where k≥2. We also show that in the space of k-dimensional C^r-maps, where k≥4 and r=2,...,∞, in any neighborhood of a map such that it has bi-focus saddle periodic orbit whose invariant manifolds are tangent, there exist open domains in which systems with flat homoclinic tangencies of corank-2 are dense (invariant manifolds forming tangency of corank-2 have a pair of linearly independent common tangent vectors).
Misato Ogawa (Ochanomizu/RIKEN): The Gibbs state on XY model
Abstract: We can reveal the statistical property of XY model by considering the Gibbs state on it. In connection with that, I'm studying about Double Variational Principle to understand the property of the Gibbs state. Then, in this presentation, I will present existence of the Gibbs state on XY model and Double Variational Principle. My presentation will extract many from others' articles. I'd like to use this place to appreciate all of their help.
Toru Sera (Osaka): TBA
Abstract: TBA
Tomoharu Suda (RIKEN): Applications of the axiomatic theory of ordinary differential equations
Abstract: In applications, we often need to generalize the notion of dynamical systems to consider ones not representable in terms of flow. When handling such cases, the question of the choice of formalism naturally arises. The axiomatic theory of ordinary differential equations is a simple yet general framework to describe many different types of systems. In this poster, we review the basics of Yorke's axiomatic theory of ODE and show how it can be applied to the study of the dynamical properties of systems that are not necessarily well-behaved.
Hiroki Takahasi (Keio): On distributions of random cycles for the Gauss-Rényi map
Abstract: In [S. Suzuki, H. Takahasi, J. Math. Anal. Appl. 527 (2023) 127465] we
have introduced the notion of `random cycles' for random dynamical systems,
as an analogue of periodic orbits for deterministic dynamical systems,
and proved that suitably weighted random cycles equidistribute with respect to
the stationary measure as the periods of the cycles tend to infinity.
This is an analogue of Bowen's weighted equidistribution theorem on periodic orbits
of topologically mixing Axiom A diffeomorphisms in the random setup.
We briefly describe a strategy to extend our argument
to the Gauss-Rényi map originating in the theory of continued fractions. A main new difficulty is that the associated skew product map
is represented as a countable full shift, and so the tightness of various sequences
of measures are far from trivial.
Airi Takeuchi (Augsburg Germany): Conformal and projective transformations on integrable billiards with potentials
Abstract: This presentation focuses on the study of the integrability of two-dimensional billiards in the presence of non-constant potentials.
For free motion in the two-dimensional plane R^2, there are two types of integrable billiard systems: circular and elliptic free billiards. In the presence of potentials (such as Kepler potential and Hooke potential) defined in the plane, there are various known integrable billiard systems. In our study, we illustrate how some of these integrable billiard systems are related to each other by conformal transformations. As an application, we obtain infinitely many integrable billiard systems defined in central force problems on a particular energy level. We then explain that the classical Hooke-Kepler correspondence extends to the correspondence between integrable Hooke and Kepler billiards.
As well as billiard systems in the plane, we consider systems on curved surfaces. We use the projective dynamical approach to establish the integrability of the Lagrange billiard systems in the plane, on the sphere, in the hyperbolic plane, defined as follows: the integrable Lagrange billiard system is defined with the Lagrange problem, which is the superposition of two Kepler problems and a Hooke problem, with the Hooke center at the middle of the Kepler centers, as the underlying mechanical system, and with any combinations of confocal conic sections with foci at the Kepler centers as reflection wall. This covers many previously known integrable mechanical billiards, especially the integrable Hooke, Kepler, and two-center billiards in the plane as subcases.
This presentation is based on the joint work with Lei Zhao(Augsburg).
Wei Hao Tey (IRCN Tokyo): Bifurcation of minimal attractors of random dynamical systems with bounded noise
Abstract: The importance of considering the presence of noise and uncertainty has become increasingly evident in real-world applications during the last few decades. With an assumption of bounded noise, the stationary measures are typically non-unique and supported on compact sets, allowing for a topological characterisation of the dynamical situation. The collection of trajectories with all possible noise realisations can be described at the topological level as a deterministic set-valued dynamical system.
We are interested in the bifurcation of the stationary measures of the random dynamical system, which turns out to be invariant sets of the set-valued system. The bifurcation can be detected by traditional bifurcation of a single-valued dynamical system, inspired by the boundary of the invariant sets.
We also present a possible early warning indicator for critical transitions of the corresponding random dynamical system with bounded noise. By extracting the tail of the stationary distribution, one can approximate the eigenvalue of the extremal map, indicating the distance from the bifurcation.
Natsuki Tsutsumi (Tokyo Marine Science Tech.): Constructing data-driven ODEs of a chaotic fluid flow
Abstract: We construct ODEs describing the macroscopic variables of a chaotic fluid flow from time-series data.
With the method to make ODEs from scalar time-series data, we construct a system of ODEs that predicts the high-frequency energy time series of the fluid flow from the low-frequency energy time series.
The constructed system is evaluated by not only short time series but also the invariant sets.
Takayuki Watanabe (Chubu): On the stochastic bifurcation of random holomorphic dynamical systems
Abstract: We consider the dynamics of randomized Newton methods from the viewpoint of stochastic bifurcation. Through numerical experiments, we show that randomized algorithms may work successfully with much smaller noise than is known in theory. As a complement to this numerical result, we also give mathematical results on randomized dynamical systems of quadratic polynomials.
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