Masato TSUJII webpage (publication)

Status: preprint

Abstract: For any C^r contact Anosov flow with r >= 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 and the hyperbolicity exponents of the flow. 

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Status:  published in "Probabilistic and Geometric Structures in Dynamics", K. Burns, D. Dolgopyat and Ya. Pesin (eds), Contemp. Math. (Amer. Math. Soc.), Volume in honour of M. Brin's 60th birthday

Abstract: 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.

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Status: Published in Ergodic Theory and Dynamical systems28
no 1, 291--317 (2008)

Abstract: We consider suspension semi-flows of angle-multiplying maps on the circle for $C^r$ ceiling functions with $r¥ge 3$. Under a $C^r$generic condition on the ceiling function, we show that there exists a Hilbert space (anisotropic Sobolev space) contained in the $L^2$ space such that the　Perron-Frobenius operator for the time-$t$-map acts naturally 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. Furthermore, the Perron-Frobenius operator for the time-$t$-map is quasi-compact for a $C^r$ open and dense set of ceiling functions.

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Status: Published in Ann. Inst. Fourier (Grenoble) 57 (2007), no. 1, 127--154.

Abstract: 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.

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Status: Published in Nonlinearity 19 (2006), no. 10, 2467--2473.

Abstract: 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.

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Status: Published in Discrete and continuous dynamical systems 15 (2006), no. 1, 21--35.

Abstract: 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, ¥lambda y+f(x)) $ where $¥ell¥geq 2$, $0<¥lambda<1$ and $f$ is a $C^r$ function on $S^1$. We show that, if $¥lambda^{1+2s}¥ell>1$ 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) $f$.

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Status: Published in Acta Math. 194, no. 1, 37--132. (2005)

Abstract: 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 $r¥ge 19$.

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