arXiv subjects
Publications and source records attributed to Masato Tsujii.
This paper is about spectral properties of transfer operators for contact Anosov flows. The main result gives the essential spectral radius of the transfer operators acting on the so-called anisotropic Sobolev space exactly in terms of dynamical exponents. Also we provide a simplified proof by using the FBI transform.
For any $C^r$ contact Anosov flow with $r\ge 3$, we construct a scale of Hilbert spaces, which are embedded in the space of distributions on the phase space and contain all $C^r$ functions, such that the transfer operators for the flow extend to them boundedly and that the extensions are quasi-compact. Further we give explicit bounds on the essential spectral radii of the extensions in terms of the differentiability r and the hyperbolicity exponents of the flow.
For smooth hyperbolic dynamical systems and smooth weights, we relate Ruelle transfer operators with dynamical Fredholm determinants and dynamical zeta functions: First, we establish bounds for the essential spectral radii of the transfer operator on new spaces of anisotropic distributions, improving our previous results. Then we give a new proof of Kitaev's lower bound for the radius of convergence of the dynamical Fredholm determinant. In addition we show that the zeroes of the determinant in the corresponding disc are in bijection with the eigenvalues of the transfer operator on our spaces of anisotropic distributions, closing a question which remained open for a decade.
We consider suspension semi-flows of angle-multiplying maps on the circle. Under a $C^r$generic condition on the ceiling function, we show that there exists an anisotropic Sobolev space contained in the $L^2$ space such that the Perron-Frobenius operator for the time-$t$-map acts on it and that the essential spectral radius of that action is bounded by the square root of the inverse of the minimum expansion rate. This leads to a precise description on decay of correlations and extends the result of M. Pollicott.
In this brief note we present a very simple strategy to investigate dynamical determinants for uniformly hyperbolic systems. The construction builds on the recent introduction of suitable functional spaces which allow to transform simple heuristic arguments in rigorous ones. Although the results so obtained are not exactly optimal the straightforwardness of the argument makes it noticeable.
This note is about the spectral properties of transfer operators associated to smooth hyperbolic dynamics. In the first two sections (written in 2006), we state our new results relating such spectra with dynamical determinants, first announced at the conference ``Traces in Geometry, Number Theory and Quantum Fields" at the Max Planck Institute, Bonn, October 2005. In the last two sections, we give a reader-friendly presentation of some key ideas in our work in the simplest possible settings, including a new proof of a result of Ruelle on expanding endomorphisms. (These last two sections are a revised version of the lecture notes given during the workshop ``Resonances and Periodic Orbits: Spectrum and Zeta functions in Quantum and Classical Chaos" at Institut Henri Poincaré, Paris, July 2005.) (Revised version, submitted for publication)
(Revised version, January 2006. S. Gouezel pointed out that, when 1<r<2, the proof in the previous version was incomplete. In fixing this gap, we simplified the argument in Section 6. In addition, there is a new appendix, with an alternative description of the norms. We also corrected a few minor flaws.) We study spectral properties of transfer operators for diffeomorphisms T on a Riemannian manifold: Suppose that there is an isolated hyperbolic subset for T, with a compact isolating neighborhood V. We first introduce Banach spaces of distributions supported on V, which are anisotropic versions of the usual space of C^p functions C^p(V) and of the generalized Sobolev spaces W^{p,t}(V), respectively. Then we show that the transfer operators associated to T and a smooth weight extend boundedly to these spaces, and we give bounds on the essential spectral radii of such extensions in terms of hyperbolicity exponents. These bounds shed some light on those obtained by Kitaev for the radius of convergence of dynamical determinants.
We consider dynamical systems generated by skew products of affine contractions on the real line over angle-multiplying maps on the circle $S^1$: $ T:S^{1}\times \R\to S^1\times \R, T(x,y)=(\ell x, λy+f(x)) $ where $\ell\geq 2$, $0<λ<1$ and $f$ is a $C^r$ function on $S^1$. We show that, if $λ^{1+2s}\ell>1$ for some $0\leq s< r-2$, the density of the SBR measure for $T$ is contained in the Sobolev space $W^s(S^1\times \R)$ for almost all ($C^r$ generic, at least) $f$.
We consider dynamical systems generated by partially hyperbolic surface endomorphisms of class C^r with one-dimensional strongly unstable subbundle. As the main result, we prove that such a dynamical system generically admits finitely many ergodic physical measures whose union of basins of attraction has total Lebesgue measure, provided that r>= 19.