arXiv subjects

Takayuki Watanabe

Publications and source records attributed to Takayuki Watanabe.

13 recordsLinked to original sources

arXiv2026

On the topology of the limit sets of non-autonomous iterated function systems

Since Mandelbrot's seminal work, there has been growing interest in the geometric nature of fractals. While the topological properties of the limit sets of IFSs have been studied -- notably in the pioneering work of Hata -- many aspects remain poorly understood, specially in the non-autonomous setting. In this paper, we investigate the topology of limit sets arising from randomly generated non-autonomous IFSs. To this end, we develop a simplicial-homological framework that makes their topological structure accessible to rigorous analysis. We apply our abstract theory to the concrete analysis of the so-called fractal squares, and provide an answer to a variant of Mandelbrot's percolation problem. Moreover, for the non-autonomous fractal squares considered here, we prove that the Betti numbers of the finite-stage approximations grow exponentially at a rate equal to the natural symbolic entropy of the system. This reveals a quantitative link between topology across scales and dynamical complexity.

math.DS↗

arXiv2026

One-shot prediction of noise-induced bifurcations with reservoir computing

Dynamical systems can exhibit complex responses when noise is injected. In particular, dynamics can be qualitatively altered by dynamic noise, a phenomenon known as noise-induced bifurcation. Predicting noise-induced bifurcations is a critical challenge in nonlinear physics. Recently, it has been reported that reservoir computing, a machine learning framework, can reconstruct the unseen global structure of a dynamical system, including bifurcations, from limited time series data. However, learning global structures in random dynamical systems has not yet been systematically addressed. In this study, we report that a simple reservoir computing framework can predict the noise-induced bifurcation structure from the time series at a single noise condition. We demonstrate dynamic noise cancellation and the reconstruction of entire noise-induced bifurcation structures, including noise-induced chaos and noise-induced order, in representative dynamical systems. Additionally, we provide a theoretical explanation for noise cancellation and demonstrate noise cancellation of a neuromorphic spintronics device. Our results provide significant insights into understanding and harnessing real-world noisy complex dynamics.

nlin.CD↗

arXiv2026

Topology of slices through the Sierpiński tetrahedron

We investigate slices of the Sierpiński tetrahedron from a topological viewpoint. For each $c\in[0,1]$, we study the Čech (co)homology group of the slice at height $c$. We show that the topology of the slice exhibits a sharp dichotomy. If $c$ is a dyadic rational, then the slice has finitely many connected components, infinite first Čech homology, and trivial higher homology. If $c$ is not a dyadic rational, then the slice is totally disconnected and all positive-degree Čech homology groups vanish.

math.DS↗

arXiv2024

Bowen's formula for a rational graph-directed Markov system

We establish Bowen's formula for the Julia set of a non-elementary, expanding, irreducible and aperiodic rational graph-directed Markov system satisfying the backward separating condition. Towards this end, we shall prove that the associated skew product map is topologically exact on the skew product Julia set, and satisfies the density of repelling periodic points. Moreover, we give a criterion for expandingness in terms of hyperbolicity.

math.DS↗

arXiv2024

On the stochastic bifurcations regarding random iterations of polynomials of the form $z^{2} + c_{n}$

In this paper, we consider random iterations of polynomial maps $z^2 +c_n$ where $c_n$ are complex-valued independent random variables following the uniform distribution on the closed disk with center $c$ and radius $r$. The aim of this paper is twofold. First, we study the (dis)connectedness of random Julia sets. Here, we reveal the relationships between the bifurcation radius and connectedness of random Julia sets. Second, we investigate the bifurcation of our random iterations and give quantitative estimates of bifurcation parameters. In particular, we prove that for the central parameter $c = -1$, almost every random Julia set is totally disconnected with much smaller radial parameters $r$ than expected. We also introduce several open questions worth discussing.

math.DS↗

arXiv2024

Backtracking New Q-Newton's method, Newton's flow, Voronoi's diagram and Stochastic root finding

math.OC↗

arXiv2023

Backtracking New Q-Newton's method, Schröder's theorem, and Linear Conjugacy

A new variant of Newton's method - named Backtracking New Q-Newton's method (BNQN) - which has strong theoretical guarantee, is easy to implement, and has good experimental performance, was recently introduced by the third author. Experiments performed previously showed some remarkable properties of the basins of attractions for finding roots of polynomials and meromorphic functions using BNQN. In particular, it seems that for finding roots of polynomials of degree 2, the basins of attraction of the dynamics for BNQN are the same as that for Newton's method (the latter is the classical Schröder's result in Complex Dynamics). In this paper, we show that indeed the picture we obtain when finding roots of polynomials of degree 2 is the same as that in Schöder's result, with a remarkable difference: on the boundary line of the basins, the dynamics of Newton's method is chaotic, while the dynamics of BNQN is more smooth. On the way to proving the result, we show that BNQN (in any dimension) is invariant under conjugation by linear operators of the form $A=cR$, where $R$ is unitary and $c>0$ a constant. This again illustrates the similarity-difference relation between BNQN and Newton's method.

math.DS↗

arXiv2022

Non-i.i.d. random holomorphic dynamical systems and the generic dichotomy

We consider non-i.i.d. random holomorphic dynamical systems whose choice of maps depends on Markovian rules. We show that generically, such a system is mean stable or chaotic with full Julia set. If a system is mean stable, then the Lyapunov exponent is uniformly negative for every initial value and almost every random orbit. Moreover, we consider families of random holomorphic dynamical systems and show that the set of mean stable systems has full measure under certain conditions. The latter is a new result even for i.i.d. random dynamical systems.

math.DS↗

arXiv2019

Non-i.i.d. random holomorphic dynamical systems and the probability of tending to infinity

We consider random holomorphic dynamical systems on the Riemann sphere whose choices of maps are related to Markov chains. Our motivation is to generalize the facts which hold in i.i.d. random holomorphic dynamical systems. In particular, we focus on the function $T$ which represents the probability of tending to infinity. We show some sufficient conditions which make $T$ continuous on the whole space and we characterize the Julia sets in terms of the function $T$ under certain assumptions.

math.DS↗

arXiv2011

Phase Separation of Multi-Component Bose-Einstein Condensates of Trapped Atoms and Molecules with a Homonuclear Feshbach Resonance

We investigate phase separation of Bose-Einstein condensates (BECs) of two-component atoms and one-component molecules with a homonuclear Feshbach resonance. We develop a full model for dilute atomic and molecular gases including correlation of the Feshbach resonance and all kinds of interparticle interactions, and numerically calculate order parameters of the BECs in spherical harmonic oscillator traps at zero temperature with the Bogoliubov's classical field approximation. As a result, we find out that the Feshbach resonance can induce two types of phase separation. The actual phase structures and density profiles of the trapped gases are predicted in the whole parameter region, from the atom dominant regime to the molecule dominant regime. We focus on the role of the molecules in the phase separation. Especially in the atom dominant regime, the role of the molecules is described through effective interactions derived from our model. Furthermore we show that a perturbative and semi-classical limit of our model reproduces the conventional atomic BEC (single-channel) model.

cond-mat.quant-gas↗

arXiv2010

Observation of Amplified Stimulated Terahertz Emission from Optically Pumped Epitaxial Graphene Heterostructures

We experimentally observe the fast relaxation and relatively slow recombination dynamics of photogenerated electrons/holes in an epitaxial graphene-on-Si heterostructure under pumping with a 1550-nm, 80-fs pulsed fiber laser beam and probing with the corresponding terahertz (THz) beam generated by and synchronized with the pumping laser. The time-resolved electric-field intensity originating from the coherent terahertz photon emission is electro-optically sampled in total-reflection geometry. The Fourier spectrum from 1.8 to 5.2 THz agrees well the pumping photon spectrum. This result is attributed to amplified emission of THz radiation from the graphene sample stimulated by the THz probe beam, and provides evidence for the occurrence of negative dynamic conductivity in the terahertz spectral range.

cond-mat.mtrl-sci↗

arXiv2007

Bose-Fermi Pair Correlations in Attractively Interacting Bose-Fermi Atomic Mixtures

We study static properties of attractively interacting Bose-Fermi mixtures of uniform atomic gases at zero temperature. Using Green's function formalism we calculate boson-fermion scattering amplitude and fermion self-energy in the medium to lowest order of the hole line expansion. We study ground state energy and pressure as functions of the scattering length for a few values of the boson-fermion mass ratio $m_b/m_f$ and the number ratio $N_b/N_f$. We find that the attractive contribution to energy is greatly enhanced for small values of the mass ratio. We study the role of the Bose-Fermi pair correlations in the mixture by calculating the pole of the boson-fermion scattering amplitude in the medium. The pole shows a standard quasiparticle dispersion for a Bose-Fermi pair, for $m_b/m_f\geq 1$. For small values of the mass ratio, on the other hand, a Bose-Fermi pair with a finite center-of-mass momentum experiences a strong attraction, implying large medium effects. In addition, we also study the fermion dispersion relation. We find two dispersion branches with the possibility of the avoided crossings. This strongly depends on the number rario $N_b/N_f$.

cond-mat.other↗

arXiv1997

First Results of Tokyo Dark Matter Search with a Lithium Fluoride Bolometer

The First results of the Tokyo dark matter search programme using a 21-g lithium fluoride bolometer are presented. The background spectrum was measured in the surface laboratory. We derive an exclusion plot for the spin-dependently coupled Weakly Interacting Massive Particles (WIMPs) cross section.

hep-ex↗