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Masato Tsujii

Publications and source records attributed to Masato Tsujii.

At least 19 recordsLinked to original sources
arXiv2026

Perturbation of the time-1 map of a generic volume-preserving $3$-dimensional Anosov flow

Let $s > 1$ be a large integer, and let $f$ be a diffeomorphism sufficiently close in the $C^{s}$-topology to the time-1 map of a $C^{s}$ generic volume-preserving Anosov flow on a $3$-dimensional compact manifold. We show that for any probability measure $\mu$ with smooth density, $f^n_* \mu$ converges exponentially fast to a common limit measure with full support. As corollaries, we show the following: $f$ is topologically mixing; $f$ has a unique physical measure with basin of full Lebesgue measure, which is also the unique u-Gibbs state; if $f$ is volume preserving, then $f$ is exponentially mixing with respect to the volume form. As applications, we give a class of time-1 maps of transitive Anosov flows non-approximable in $C^{s}$ by Axiom A maps, giving negative answer to a question of Palis-Pugh (1974); the first example of a $C^{s}$-stably transitive time-1 map of Anosov flow, a question mentioned in Bonatti-Guelman (2010), Rodriguez Hertz (2010); as well as the first example of a $C^{s}$-stably transitive diffeomorphism without periodic points.

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arXiv2026

Non-stationary normal coordinates on neighborhoods of Pesin stable manifolds

We construct non-stationary normal coordinates in a neighborhood of Pesin stable manifolds \cite{Pesin}. This construction is a natural extension, via jets in the normal directions, of the non-stationary normal coordinates on stable manifolds introduced by Guysinsky and Katok. We emphasize that this extension provides a useful framework for describing the fractal geometric structures of stable and unstable foliations in smooth dynamical systems.

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arXiv2024

Polynomial rate of mixing for the heterochaos baker maps with mostly neutral center

For the heterochaos baker maps whose central direction is mostly neutral, we prove that correlations for H\"older continuous functions decay at an optimal polynomial rate of order $n^{-3/2}$. Our method of proof relies on a description of the action of a reduced Perron-Frobenius operator by means of a comparison to the symmetric simple random walk with an absorbing wall, aka `gambler's ruin problem'.

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arXiv2023

Fractal Weyl law for the Ruelle spectrum of Anosov flows

On a closed manifold $M$, we consider a smooth vector field $X$ that generates an Anosov flow. Let $V\in C^{\infty}\left(M;\mathbb{R}\right)$ be a smooth function called potential. It is known that for any $C>0$, there exists some anisotropic Sobolev space $\mathcal{H}_{C}$ such that the operator $A=-X+V$ has intrinsic discrete spectrum on $\mathrm{Re}\left(z\right)>-C$ called Ruelle resonances. In this paper, we show a fractal Weyl law: the density of resonances is bounded by $O\left(\left\langle ω\right\rangle ^{\frac{n}{1+β_{0}}}\right)$ where $ω=\mathrm{Im}\left(z\right)$, $n=\mathrm{dim}M-1$ and $0<β_{0}\leq1$ is the Hölder exponent of the distribution $E_{u}\oplus E_{s}$ (strong stable and unstable). We also obtain some more precise results concerning the wave front set of the resonances and the invertibility of the transfer operator. Since the dynamical distributions $E_{u}$,$E_{s}$ are non smooth, we use some semi-classical analysis based on wave packet transform associated to an adapted metric $g$ on $T^{*}M$ and construct some specific anisotropic Sobolev spaces.

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arXiv2022

Virtually expanding dynamics

We introduce a class of discrete dynamical systems that we call \emph{virtually expanding}. This is an open subset of self-covering maps on a closed manifold which contains all expanding maps and some partially hyperbolic volume-expanding maps. We show that the Perron-Frobenius operator is quasi-compact on a Sobolev space of positive order for such class of dynamical systems.

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arXiv2021

Invariant densities for random continued fractions

We continue the study of random continued fraction expansions, generated by random application of the Gauss and the Rényi backward continued fraction maps. We show that this random dynamical system admits a unique absolutely continuous invariant measure with smooth density.

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arXiv2021

Micro-local analysis of contact Anosov flows and band structure of the Ruelle spectrum

We develop a geometrical micro-local analysis of contact Anosov flow, such as geodesic flow on negatively curved manifold. This micro-local analysis is based on wave-packet transform discussed in arXiv:1706.09307. The main result is that the transfer operator is well approximated (in the high frequency limit) by the quantization of the Hamiltonian flow naturally defined from the contact Anosov flow and extended to some vector bundle over the symplectization set. This gives a few important consequences: the discrete eigenvalues of the generator of transfer operators, called Ruelle spectrum, are structured into vertical bands. If the right-most band is isolated from the others, most of the Ruelle spectrum in it concentrate along a line parallel to the imaginary axis and, further, the density satisfies a Weyl law as the imaginary part tend to infinity. Some of these results were announced in arXiv:1301.5525.

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arXiv2018

On cohomological theory of dynamical zeta functions

We discuss about the conjectural cohomological theory of dynamical zeta functions in the case of general Anosov flows. Our aim is to provide a functional-analytic framework that enables us to justify the basic part of the theory rigorously. We show that the zeros and poles of a class of dynamical zeta functions, including the semi-classical (or Gutzwiller-Voros) zeta functions, are interpreted as eigenvalues of the generators of some transfer operators acting on the leaf-wise de Rham cohomology spaces of the unstable foliation.

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arXiv2016

The semiclassical zeta function for geodesic flows on negatively curved manifolds

We consider the semi-classical (or Gutzwiller-Voros) zeta function for $C^\infty$ contact Anosov flows. Analyzing the spectrum of transfer operators associated to the flow, we prove, for any $τ>0$, that its zeros are contained in the union of the $τ$-neighborhood of the imaginary axis, $|\Re(s)|<τ$, and the region $\Re(s)<-χ_0+τ$, up to finitely many exceptions, where $χ_0>0$ is the hyperbolicity exponent of the flow. Further we show that the zeros in the neighborhood of the imaginary axis satisfy an analogue of the Weyl law.

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arXiv2016

The partial captivity condition for U(1) extensions of expanding maps on the circle

This paper concerns the compact group extension \[ f:\mathbb{T}^2\to \mathbb{T}^2,\quad f (x,s)= (E(x), s+τ(x)\ \text{mod }1) \] of an expanding map $E:\mathbb{S}^1\to \mathbb{S}^1$. The dynamics of $f$ and its stochastic perturbations have previously been studied under the so-called partial captivity condition. Here we prove a supplementary result that shows that partial captivity is a $\mathscr{C}^r$ generic condition on $τ$, once we fix $E$.

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arXiv2015

The error term of the prime orbit theorem for expanding semiflows

We consider suspension semiflows of an angle multiplying map on the circle and study the distributions of periods of their periodic orbits. Under generic conditions on the roof function, we give an asymptotic formula on the number $π(T)$ of prime periodic orbits with period $\le T$. The error term is bounded, at least, by \[ \exp((1-\frac{1}{4\lceil χ_{\max}/h_{\mathrm{top}}\rceil}+\varepsilon) h_{\top} T)\qquad {in the limit $T\to \infty$} \] for arbitrarily small $\varepsilon>0$, where $h_{\mathrm{top}}$ and $χ_{\max}$ are respectively the topological entropy and the maximal Lyapunov exponent of the semiflow.

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arXiv2013

Band structure of the Ruelle spectrum of contact Anosov flows

If X is a contact Anosov vector field on a smooth compact manifold M and V is a smooth function on M, it is known that the differential operator A=-X+V has some discrete spectrum called Ruelle-Pollicott resonances in specific Sobolev spaces. We show that for |Im(z)| large the eigenvalues of A are restricted to vertical bands and in the gaps between the bands, the resolvent of A is bounded uniformly with respect to |Im(z)|. In each isolated band the density of eigenvalues is given by the Weyl law. In the first band, most of the eigenvalues concentrate of the vertical line Re(z)=< D >, the space average of the function D(x)=V(x)-1/2 div(X)/E_u where Eu is the unstable distribution. This band spectrum gives an asymptotic expansion for dynamical correlation functions.

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arXiv2013

Prequantum transfer operator for symplectic Anosov diffeomorphism

We define the prequantization of a symplectic Anosov diffeomorphism f:M-> M, which is a U(1) extension of the diffeomorphism f preserving an associated specific connection, and study the spectral properties of the associated transfer operator, called prequantum transfer operator. This is a model for the transfer operators associated to geodesic flows on negatively curved manifolds (or contact Anosov flows). We restrict the prequantum transfer operator to the N-th Fourier mode with respect to the U(1) action and investigate the spectral property in the limit N->infinity, regarding the transfer operator as a Fourier integral operator and using semi-classical analysis. In the main result, we show a " band structure " of the spectrum, that is, the spectrum is contained in a few separated annuli and a disk concentric at the origin. We show that, with the special (Hölder continuous) potential V0=1/2 log |det Df_x|_{E_u}|, the outermost annulus is the unit circle and separated from the other parts. For this, we use an extension of the transfer operator to the Grassmanian bundle. Using Atiyah-Bott trace formula, we establish the Gutzwiller trace formula with exponentially small reminder for large time. We show also that, for a potential V such that the outermost annulus is separated from the other parts, most of the eigenvalues in the outermost annulus concentrate on a circle of radius exp where <.> denotes the spatial average on M. The number of these eigenvalues is given by the "Weyl law", that is, N^d.Vol(M) with d=1/2. dim(M) in the leading order. We develop a semiclassical calculus associated to the prequantum operator by defining quantization of observables Op(psi) in an intrinsic way. We obtain that the semiclassical Egorov formula of quantum transport is exact. We interpret all these results from a physical point of view as the emergence of quantum dynamics in the classical correlation functions for large time. We compare these results with standard quantization (geometric quantization) in quantum chaos.

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arXiv2013

On the Fourier transforms of self-similar measures

For the Fourier transform $\mathcal{F}μ$ of a general (non-trivial) self-similar measure $μ$ on the real line $\mathbb{R}$, we prove a large deviation estimate \[ \lim_{c\to +0} \varlimsup_{t\to \infty}\frac{1}{t}\log (\mathrm{Leb}\{x\in [-e^t, e^t]\mid |\mathcal{F}μ(ξ)| \ge e^{-ct} \})=0. \]

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