Titles and abstracts of talks
The symbols # in front of titles indicate short communications.
10th (Fri) Chairs: Hiroki Takahasi (Keio University) AM; Yutaka Ishii (Kyushu University) PM
10:05-10:50 Shota Osada (Kagoshima University): On the ergodicity of unlabeled dynamics associated with random point fields
Abstract: We consider a reversible diffusion on the configuation space over $\mathbb{R}^d$. Under a mild condition, we show that, for a random point field $\mu$, $\mu$-reversible diffusion is ergodic under the time-shift of a path iff $\mu$ is $\mathscr{G}_{\infty}$-trivial. We also present some applications. This is joint work with H. Osada (Chubu university).
11:00-11:45 Yong Moo Chung (Hiroshima University): On the criterion of the Erd¥H{o}s-R¥'{e}nyi law for intermittent maps
Abstract: It is well known that the stochastic properties of dynamical systems modeled by Young towers depend on the return time functions to the base sets.
In this talk, we discuss from this point of view a sufficient condition for the Erd¥H{o}s-R¥'{e}nyi law to hold for a class of intermittent maps.
14:00-14:45 Yuto Nakajima (Tokai University): Arithmetic progressions in digits in the continued fraction
Abstract: We consider an irrational number for which the sequence
of digits in its regular continued fraction expansion defines an
injection on the set of natural numbers. We further impose some restrictions on appearance of digits and determine the Hausdorff dimension of the set of such irrationals for which sequences of digits contain arbitrarily long arithmetic progressions. This is a joint work with Hiroki Takahasi (Keio University).
15:00-15:45 Nima Alibabaei (Kyoto University): Analysis of matrix product and dimension of sofic sets
Abstract: In this talk, I will discuss the dimension theory of sofic affine-invariant fractals. Previous research shows that the dimension is expressed as a limit involving complicated matrix products. However, this formula is often too complex to be understood in isolation. By analyzing these matrix products, I will demonstrate how the dimension of certain sofic sets can be viewed as the root of an equation of infinite degree.
15:45-16:00 Mao Shinoda (Ochanomizu University): #Rate distortion dimension of the Gibbs measures on the XY-model
Abstract: In this talk we consider the XY-model and a potential that depends only on the first two coordinates. We then calculate the rate distortion dimension of the Gibbs measure of the potential and discuss the relation with the mean dimension with a potential. This is a joint work with Kanji Inui.
16:15-17:00 Yuki Takahashi (Saitama University): Absolute continuity of self-similar measures for Iterated Function Systems with inverses
Abstract: We consider parameter families of self-similar measures of homogeneous Iterated Function Systems that contain inverses in the overlapping case, and show that if the random walk entropy is greater than the Lyapunov exponent, then the set of exceptional parameter such that the associated self-similar measure is singular has zero Hausdorff dimension.
17:15-18:00 Minsung Kim (KTH Royal Institute of Technology): Anisotropic spaces and nil-automorphisms
Abstract: Interconnections between parabolic and hyperbolic dynamics have been recently studied in terms of renormalization techniques. In particular, they were observed in the deviation of ergodic integrals and solving cohomological equations for some parabolic flows (cf. Liverani-Giulietti, Adam-Baladi, Faure-Gouëzel-Lanneau, and Butterley-Simonelli, etc.)
In this talk, we introduce geometric anisotropic Banach spaces on Heisenberg nilmanifolds. This construction shows how the Ruelle resonances for the transfer operator associated with the renormalization map (partially hyperbolic nil-automorphism) are related to the invariant distributions for cohomological equations of Heisenberg nilflows. This is joint work with Oliver Butterley.
11th (Sat) Chairs: Naoya Sumi (Kumamoto University) AM; Masato Tsujii (Kyushu University) PM
9:30-10:15 Yasushi Nagai (Shinshu University): Overlap coincidences for general S-adic tilings and a special case of the Pisot conjecture
Abstract: In the theory of tilings and tiling dynamical systems, it is important
to determine when a system has a pure point dynamical spectrum.
The classical overlap algorithm gives a necessary and sufficient condition
for pure point spectra for the class of self-affine tilings. In this talk,
we generalize this algorithm to the larger class of S-adic tilings
and give a sufficient condition for pure point spectra.
10:30-11:15 Eiko Kin (Osaka University): Agol cycles of pseudo-Anosov maps on the 2-punctured torus and 5-punctured sphere
Abstract: Given a periodic splitting sequence of a measured train track, an Agol cycle is the part that constitutes a period up to the action of a pseudo-Anosov map and the rescaling by its dilatation. We consider a family of pseudo-Anosov maps on the 2-punctured torus and on the 5-punctured sphere. We present measured train tracks and compute their Agol cycles. We give a condition under which two maps in the defined family are conjugate or not. In the process, we find a new formula for the dilatation. This is joint work with Jean-Baptiste Bellynck.
11:15-11:30 Yuta Mizutani (Osaka University): #The Dynnikov coordinate system for pseudo-Anosov braids
Abstract: I explain the Dynnikov coordinate system, which puts global coordinates on the space of measured foliations on punctured disk. I also explain some application of the Dynnikov coordinate system for computing the stretch factors of pseudo-Anosov braids.
11:30-11:45 Sogo Murakami (University of Tokyo): #Shadowing property and chain recurrent set with an attached singularity
Abstract: The theory of shadowing property has been developed in the study of uniformly hyperbolic dynamical systems. Arbieto et al. proved that chain recurrent sets with attached singularities satisfying a certain condition fail to exhibit the shadowing property. In this talk, we give an extension of this result.
14:00-14:45 Ziyu Liu (Shanghai Dianji University): Escape rate spectra for the geodesic flow on hyperbolic Z-covers of quotient surfaces of generalized Schottky groups
Abstract: Given a hyperbolic surface, we are interested in analyzing the escape rates of geodesic trajectories on it with various initial directions. For this purpose, we define escape rate spectra for the geodesic flow on a hyperbolic surface. After some discussions about the general case, we shall restrict our attention to a special class of hyperbolic surfaces, namely what we call the hyperbolic Z-covers of quotient surfaces of generalized Schottky groups. We shall present our results on the escape rate spectra for hyperbolic Z-covers of quotient surfaces of generalized Schottky groups. This talk is based on a joint work with Johannes Jaerisch (Nagoya University).
15:00-15:45 Jun Ueki (Ochanomizu University): An analogue of the classical Neukirch--Uchida theorem for a 3-manifold endowed with a Chebotarev link
Abstract: The analogy between knots and primes, or 3-manifolds and number rings has played a role from the era of Gauss.
A basic portion is the Galois theories of ramified covers. In number theory, the classical Neukirch--Uchida theorem asserts that a number field is determined
by its absolute Galois group. In its proof, the Hilbert ramification theory and the Chebotarev density theorem play important roles.
In low dimensional topology, McMullen constructed a sequence of knots obeying the Chebotarev law, and it may be seen as an analogue of the set of all prime numbers.
For a 3-manifold endowed with such a sequence, an analogue of the absolute Galois group is defined, and we may discuss a Neukirch--Uchida type theorem.
As is classically well-known, Mostow rigidity asserts that a hyperbolic 3-manifold is determined by its fundamental group. Our version asserts something different and new.
This talk is based on a joint work with Nadav Gropper (U.Penn/Haifa) and Yi Wang (U.Penn/Illinois).
16:00-16:45 Atsushi Katsuda (Kyushu University): Nilpotent Floquet-Bloch theory and asymptotics of closed geodesics
Abstract: In this talk, we present our version of the generalized Floquet-Bloch theory for discrete nilpotent groups. We demonstrate its application to various asymptotic problems related to nilpotent extensions, including heat kernels and closed geodesics. If time permits, we will also delve into the quasi-morphism version and explore its connections to modified Riemann-Hilbert problems.
17:00-17:45 Natsuki Tsutsumi: Ideal data-driven model using RfR method and its dynamical system analysis
Abstract: One of the aims of data-driven dynamical modeling is to predict short/long-term trajectories. However, the success of the numerical trajectory prediction does not always imply that the constructed model mimics the actual dynamics in terms of dynamical structures.
The RfR method[1,2,3] is a method for constructing an ODE system from scalar time series without background knowledge only using understandable variables. A constructed model by the RfR method can predict short/long-term trajectories well, even when the actual dynamics is realized on the model chaotic saddle instead of an attractor.
We construct an ideal model which reconstructs the full Lyapunov spectrum. Additionally, we investigate the geometric realization of the model attractor.
1.N.Tsutsumi, K.Nakai and Y.Saiki, Chaos 32, 091101 (2022)
2.N.Tsutsumi, K.Nakai and Y.Saiki, Phys.Rev.E 108, 054220 (2023)
3.N.Tsutsumi, K.Nakai and Y.Saiki, Phys.Rev.E, accepted for publication
17:45-18:00 Katsuki Ito (University of Tsukuba): #Self-similarity of Smith turtle tiling
Abstract: In plane geometry, the “einstein problem” concerns the existence of an aperiodic monotile (a single tile that can tile the whole plane and the resulting tiling must not have any periods). In 2023, D. Smith et al. found the answer to this problem: Hat, which allows reflection, and Spectre, which does not. A tiling by each tile is known to be self-similar, and in particular Hat tiling is closely related to the golden ratio. In this talk, I try visualizing the self-similarity of a tiling by Turtle, a variant of Hat, and describe an approach to investigate the rigidity of Turtle tiling. This is a joint work with Shigeki Akiyama and Tadahisa Hamada (University of Tsukuba).
12th (Sun) Chairs: Tomoyuki Miyaji (Kyoto University) AM; Zin Arai (Institute of Science Tokyo) PM1; Shunsaku Nii (Kyushu University) PM2
9:30-10:15 Masato Hara (Kyoto University): A mathematical mechanism of reservoir computing for dynamical systems
Abstract: Reservoir computing (RC) is a machine learning scheme using a kind of recurrent neural network, and it can be treated as a dynamical system in a natural way. We focus on the reconstruction task of a given dynamical system $f$. In this case, the reservoir seems to become conjugate with $f$ judging from many numerical examples, but the essential mechanism is not known well. In this talk, we will introduce an outline of the proof of a theorem which claims that a reservoir finally becomes semi-conjugate (in the weak sense) to $f$ under some assumptions for $f$ and for the reservoir. In particular, this theorem explains a mechanism of RC for hyperbolic toral automorphisms. This is a joint work with Professor Hiroshi Kokubu (Kyoto U).
10:30-11:15 Yuzuru Sato (Hokkaido University): Noise-induced phenomena in high dimensions
Abstract: Noise-induced phenomena are caused by interactions between deterministic dynamics and external noise. When a transition occurs owing to small noise, the stationary distribution of the deterministic dynamical system is substantially altered, and the unobservable structure of the original dynamics becomes observable. In such cases, nonlinear phenomena, which qualitatively differ from deterministic dynamics, emerge in the noised dynamics. This talk includes a brief review of classical noise-induced phenomena from random dynamical systems point of view. Recent results on multiple noise-induced transitions proved by validated numerics, and heterogeneous noise-induced phenomena in a class of high-dimensional dynamical systems are presented as well.
11:15-11:30 Koji Wada (Hokkaido University): #Cluster analysis of Globally Coupled Maps using optimal transport distance
Abstract: A mathematical model of a complex system, called Globally Coupled Maps (GCM), has been proposed.
GCM is a discrete dynamical system consisting of globally coupled logistic maps.
A feature of GCM is that clusters emerge through synchronization among elements, with the stability of these clusters depending on some parameters.
Specifically, it is known that cluster structure vary due to changes in the combination of synchronized elements within specific parameter region.
In this presentation, we use optimal transport distance to caputure cluster dynamics,
and we quantitatively evaluate orbit structure within this parameter region.
Additionally, we verify the effectiveness of this approach by comparing it with changes in effective dimension (i.e., the number of clusters).
14:00-14:45 Mitsuru Shibayama (Kyoto University): Variational construction of heteroclinic orbits in the planar Sitnikov problem
Abstract: Using the variational method, Chenciner and Montgomery (2000) proved the existence of a figure-eight orbit in the planar three-body problem with equal masses. Since then, a number of solutions to the N-body problem have been discovered.
The Sitnikov problem is a special case of the three-body problem. This system is known to be chaotic and has been studied using symbolic dynamics (J. Moser, 1973).
In this talk, we study the limiting case of the Sitnikov problem. Using the variational method, we show the existence of various kinds of solutions in the planar Sitnikov problem. For a given symbolic sequence, we demonstrate the existence of orbits realizing it. We also prove the existence of periodic orbits and heteroclinic orbits connecting them. This is joint work with Yuika Kajihara and Guowei Yu.
15:00-15:45 Noriaki Kawaguchi (Institute of Science Tokyo): Shadowing and the basins of terminal chain components
Abstract: Shadowing is an important concept in the topological theory of dynamical systems. It was derived from the study of hyperbolic differentiable dynamics and generally refers to a situation in which coarse orbits, or pseudo-orbits, can be approximated by true orbits. It is worth noting that the shadowing is known to be generic in the space of homeomorphisms or continuous self-maps of a closed differentiable manifold and so plays a significant role in the study of topologically generic dynamics. Chain components, which appear in the so-called Fundamental Theorem of Dynamical Systems by Conley, are basic objects for global understanding of dynamical systems. In this talk, we show that (1) if a continuous self-map of a compact metric space has the shadowing, then the union of the basins of terminal (attractor-like) chain components is a dense G-delta subset of the space; and (2) if a continuous self-map of a locally connected compact metric space has the shadowing, and if the chain recurrent set is totally disconnected, then the map is almost chain continuous.
16:00-16:15 Hibiki Kato (Hitotsubashi University): #Laminar chaotic saddle within a turbulent attractor
Abstract: Intermittent switchings between weakly chaotic (laminar) and strongly chaotic (bursty) states are often observed in systems with high-dimensional chaotic attractors, such as fluid turbulence. They differ from the intermittency of a low-dimensional system accompanied by the stability change of a fixed point or a periodic orbit in that the intermittency of a high-dimensional system tends to appear in a wide range of parameters. This paper considers a case where the skeleton of a laminar state $L$ exists as a proper chaotic subset $S$ of a chaotic attractor $X$ , that is, $S\subsetneq X$ . We characterize such a laminar state $L$ by a chaotic saddle $S$, which is densely filled with periodic orbits of different numbers of unstable directions. This study demonstrates the presence of chaotic saddles underlying intermittency in fluid turbulence and phase synchronization. Furthermore, we confirm that
chaotic saddles persist for a wide range of parameters. Also, a kind of phase synchronization turns out to occur in the turbulent model.
This talk is based on Kato, Kobayashi, Saiki and Yorke, Phys. Rev. E, 2024.
16:15-16:30 Shu Sakaguchi (Kyoto University): #The dynamics of linkages associated with Anosov flows
Abstract: A linkage is a series of rigid rods with fixed pivots and movable joints. Its motion can be viewed as a geodesic flow in the configuration space with the metric induced from the kinetic energy. By this correspondence, Hunt and Mackay (2003) showed that the free motion of the triple linkage of Thurston and Weeks is Anosov with certain parameters via the curvature of the configuration space. In this talk, we consider the deformation of the triple linkage into three dimensions. We calculate the curvature of the configuration space of the linkage and confirmed that the linkage is also Anosov with a proper constraint. This is a joint work with Mitsuru Shibayama.
16:30-16:45 Ryutaro Ichikawa (Kyoto University): #The existence of invariant curves in Origami maps
Abstract: In the origami crease-pattern, it is known that periodically arranged modules composed of triangles form a cylindrical shape, and the surfaces formed often exhibit non-trivial behaviors, which has been studied. In this study, for discrete dynamical systems formulated on origami tessellations forming cylindrical shapes, we theoretically prove the existence of invariant curves and invariant attractors when the number of modules is sufficiently large, using the Comformally KAM theorem. In addition, numerical experiments have confirmed that the invariant curves remain in the system when the number of modules is actually varied in a concrete example.
17:00-17:15 Donggeon Kim (Kyoto University): #Feedback control of the Kuramoto model with natural frequencies defined on uniform graphs
Abstract: We study feedback control of the Kuramoto model (KM) with natural frequencies
on a uniform graph which may be complete simple, random dense or random
sparse. We choose as the target orbit the synchronized state in which all
oscillators rotate with the same rotational speed, and design the controller
using the continuum limit (CL). When the graph is complete simple, we prove
that if the feedback gain is larger than a critical value, then there exists an
asymptotically stable synchronized solution that tends to the target orbit
as the feedback gain goes to infinity, and that the CL has an asymptotically
stable continuous solution which corresponds to the asymptotically stable
solution to the KM. When the graph is random, we show that the continuous
solution to the same CL as in the above case behaves as an asymptotically
stable one in the KM. We demonstrate the theoretical results by numerical
simulations for the KM on the three types of graphs. This is a joint work with Kazuyuki Yagasaki (Kyoto University).
17:15-17:30 Fumihiko Nakamura (Kitami Institute of Technology): #The shape of Young tableau generated by deterministic system
Abstract: The problem of the asymptotic behavior of the longest increasing subsequence of permutations in the symmetric group $S_n$ as $n\to\infty$ is known as the Ulam–Hammersley problem. The approach using Young tableaux, constructed through a certain algorithm and sequences generated by i.i.d. random variables, is often employed to address this problem. In this talk, we consider the limit shape of Young tableaux constructed by deterministic maps. This is a joint work in progress with Yushi Nakano (Hokkaido University) and Shinsuke Iwao (Keio University).
17:30-17:45 Hajime Tateishi (Kyoto University): #Halpern's billiard table and its invariant curves
Abstract: For a planar closed convex billiard table, it is known that the billiard map is an area-preserving twist map.
A billiard map has infinitely many invariant curves near the boundary if the boundary curve is sufficiently smooth.
On the other hand, as a condition where invariant curves do not exist in some regions, it is known that the curvature vanishes at some points on the boundary or the curvature has discontinuity.
In this talk, we consider Halpern's billiard table, which has a billiard trajectory where a ball converges at a point on the boundary, and we show it has no invariant curves near the boundary.
17:45-18:00 Eran Igra (Shanghai Institute of Mathematics and Interdisciplinary Sciences): #Essential dynamics in chaotic attractors
Abstract: Assume we have a surface diffeomorphism which can be deformed by some relative isotopy to a Pseudo-Anosov map. It is well-known that as we return along the isotopy to the original map all the periodic orbits for the Pseudo-Anosov map persist – or in other words, the topology forces the existence of complex dynamics which cannot be removed under relative isotopies (in particular, the Pseudo-Anosov map is “dynamically minimal”). Now, let us replace our surface diffeomorphism with a smooth flow on a three-dimensional manifold – then, can we find some topological constrain on the flow which forces complex dynamics to appear? For example, a constrain which forces the existence of a strange chaotic attractor?
In this talk we give a partial answer to this question. Inspired by the Thurston-Nielsen Classification Theorem and the Orbit Index Theory we prove that in certain heteroclinic scenarios one can define a class of periodic orbits for the flow which persist – without changing their knot type – under a certain class of smooth homotopies of the vector field which keep the heteroclinic condition fixed. This has the following meaning: assume we can smoothly deform the dynamics trapped between a heteroclinic knot into a hyperbolic (or more precisely, singular hyperbolic) dynamical system, then all the periodic orbits for the singular hyperbolic system are also generated by the original flow. Following that will show how our results can be applied to study the dynamics of the Lorenz and Rössler attractors, and time permitting, we will conjecture how our can be generalized to derive a forcing theory for three-dimensional flows.
13th (Mon) Chair:
9:30-10:15 Koichi Hiraide (Osaka Metropolitan University): Characterization of algebraically invariant curves at hyperbolic fixed points via normalized iterates
Abstract: In this talk, we provide a characterization of the invariant curves, such as stable and unstable manifolds, of the Henon map at a hyperbolic fixed point using the method of normalized iterates. This gives us new holomorphic approximations of the invariant curves.
10:30-11:15 Kazuyuki Yagasaki (Kyoto University): Bifurcations in the Kuramoto model defined on graphs and its continuum limits
Abstract: In this talk, I present my recent research on bifurcations in the Kuramoto model (KM) defined on deterministic dense, random dense and random sparse graphs and its continuum limits (CLs). After reviewing my previous results with Ryosuke Ihara, we describe novel results for bifurcations of the following solutions: (i) synchronized solutions in the classical KM with deterministic natural frequencies, which is defined on the complete simple graph; (ii) synchronized solutions in the KM with random natural frequencies on uniform graphs; (iii) synchronized solutions in the KM with two-mode interaction depending on two graphs, one of which is uniform and the other is nearest neighbor; (iv) twisted solutions in the KM on nearest neighbor graphs; (v) twisted solutions in the KM with feedback control on nearest neighbor graphs. In the first result, bifurcations of synchronized solutions in the KM and CL is proven to be very different. In the last three results, the corresponding CLs are analyzed by using center manifold reduction and new stable or unstable solutions are shown to be born at the bifurcations.
11:30-12:15 Yushi Nakano (Hokkaido University): Quasi strong law of large numbers for open chaotic systems
Abstract: This talk concerns quasi (metastable/transient/conditional) limit theorems for open chaotic systems. In particular, I propose a formulation of quasi strong law of large numbers. As a by-producst of this investigation, I can also discuss about pointwise ergodic theorem for conditionally invariant probability measures, quasi law of the iterated logarithm and quasi almost sure invariant principle. This is partially based on a joint work with J. Atnip, G. Froyland, C. Gonzalez-Tokman and S. Vaienti.
12:15-12:30 Hiroki Takahasi (Keio University): #The Hausdorff dimension of recurrent sets for non-Markov shifts
For a class of non-Markov subshifts over a finite alphabet, including any beta shift, any S-gap shift and some coded shifts,
we show that any generalized level set of the recurrent set has full Hausdorff dimension.
This result considerably generalizes that of
Feng and Wu [Nonlinearity 14 (2001) 81--85] on Markov shifts. A main ingredient of a proof of the result
is a new finite approximation of ergodic measures with large entropy developped
by Shinoda, Takahasi, Yamamoto [arXiv:2406.01123] that relies on Climenhaga-Thompson's theory
on subshifts with non-uniform specification.
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