Dynamical determinants and spectrum for hyperbolic diffeomorphisms

By Viviane Baladi, M. Tsujii. 2006

97 citations recorded by the source corpus.

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.

Subfields: Geometry and Topology, Mathematical Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/b77190e73371730dd83d583c69ea3de210cae7ac

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Citations in the corpus

The semiclassical zeta function for geodesic flows on negatively curved manifolds (2013, background): “3 for completeness, based on the idea presented in [5] in the case of hyperbolic discrete dynamical systems.”
Smooth Anosov flows: Correlation spectra and stability (2007, background): “The recent papers [19, 18, 12, 2, 3, 20, 9, 7, 21, 4, 13], have shown that such an approach yields a simpler and far reaching alternative to the more traditional point of view based on Markov partitions.”
Compact locally maximal hyperbolic sets for smooth maps: fine statistical properties (2006, unknown): “A partial answer to the last question was first given in [Liv05] and a complete solution will appear in the paper [BT06].”
Anosov flows and dynamical zeta functions (2012, background): “It is possible that such methods could be adapted to the present case and that such versions might allow one to obtain sharper estimates when one has finite differentiability (as in BaladiTsujii [6]) or when one can exploit an additional contact structure (see Tsujii [46]).”