Fredholm determinants, Anosov maps and Ruelle resonances
Carlangelo Liverani
Discrete & Continuous Dynamical Systems - A 2005, 13(5): 1203-1215 doi: 10.3934/dcds.2005.13.1203
I show that the dynamical determinant, associated to an Anosov diffeomorphism, is the Fredholm determinant of the corresponding Ruelle-Perron-Frobenius transfer operator acting on appropriate Banach spaces. As a consequence it follows, for example, that the zeroes of the dynamical determinant describe the eigenvalues of the transfer operator and the Ruelle resonances and that, for $\C^\infty$ Anosov diffeomorphisms, the dynamical determinant is an entire function.
keywords: Anosov systems. zeta functions Dynamical determinants
Coupled map lattices without cluster expansion
Gerhard Keller Carlangelo Liverani
Discrete & Continuous Dynamical Systems - A 2004, 11(2&3): 325-335 doi: 10.3934/dcds.2004.11.325
We present an approach to the investigation of the statistical properties of weakly coupled map lattices that avoids completely cluster expansion techniques. Although here it is implemented on a simple case we expect similar strategies to be applicable in a much larger class of situations.
keywords: Coupled map lattice exponential decay of correlations transfer operator spatio-temporal chaos. spectral gap
A footnote on expanding maps
Carlangelo Liverani
Discrete & Continuous Dynamical Systems - A 2013, 33(8): 3741-3751 doi: 10.3934/dcds.2013.33.3741
I introduce Banach spaces on which it is possible to precisely characterize the spectrum of the transfer operator associated to a piecewise expanding map with Hölder weight.
keywords: decay of correlations transfer operator. Expanding maps
Xavier Bressaud Y. Lacroix Carlangelo Liverani Sandro Vaienti
Discrete & Continuous Dynamical Systems - A 2006, 15(1): i-ii doi: 10.3934/dcds.2006.15.1i
As the field of ergodic theory has branched out in considerably many directions it has become harder, even for the expert, to keep a unitary vision of the field.
   From May 17th, 2004 to May 28th, 2004 the following were held in Marseille, CIRM; first a one week school on ergodic theory devoted to the presentation of several topics of the field, second, a one week conference focused on the recent advances in the study of non-uniformly hyperbolic dynamical systems, in connection with smooth ergodic theory.
   The idea then emerged of the edition of a special issue collecting contributions to this two week event, providing a vast panorama on ergodic theory especially addressed to people working in concrete smooth dynamical systems.

For more information please click the “Full Text” above.
keywords: dynamics.
Smooth Anosov flows: Correlation spectra and stability
Oliver Butterley Carlangelo Liverani
Journal of Modern Dynamics 2007, 1(2): 301-322 doi: 10.3934/jmd.2007.1.301
By introducing appropriate Banach spaces one can study the spectral properties of the generator of the semigroup defined by an Anosov flow. Consequently, it is possible to easily obtain sharp results on the Ruelle resonances and the differentiability of the SRB measure.
keywords: Transfer operator resonances differentiability SRB.
On the work and vision of Dmitry Dolgopyat
Carlangelo Liverani
Journal of Modern Dynamics 2010, 4(2): 211-225 doi: 10.3934/jmd.2010.4.211
We present some of the results and techniques due to Dolgopyat. The presentation avoids technicalities as much as possible while trying to focus on the basic ideas. We also try to present Dolgopyat's work in the context of a research program aimed at enlightening the relations between dynamical systems and nonequilibrium statistical mechanics.
keywords: Brin prize Dolgopyat.
Robustly invariant sets in fiber contracting bundle flows
Oliver Butterley Carlangelo Liverani
Journal of Modern Dynamics 2013, 7(2): 255-267 doi: 10.3934/jmd.2013.7.255
We provide abstract conditions which imply the existence of a robustly invariant neighborhood of a global section of a fiber bundle flow. We then apply such a result to the bundle flow generated by an Anosov flow when the fiber is the space of jets (which are described by local manifolds). As a consequence we obtain sets of manifolds (e.g., approximations of stable manifolds) that are left invariant for all negative times by the flow and its small perturbations. Finally, we show that the latter result can be used to easily fix a mistake recently uncovered in the paper Smooth Anosov flows: correlation spectra and stability [2] by the present authors.
keywords: Transfer operators resonances differentiability of SRB. Bundle flows
Quenched CLT for random toral automorphism
Arvind Ayyer Carlangelo Liverani Mikko Stenlund
Discrete & Continuous Dynamical Systems - A 2009, 24(2): 331-348 doi: 10.3934/dcds.2009.24.331
We establish a quenched Central Limit Theorem (CLT) for a smooth observable of random sequences of iterated linear hyperbolic maps on the torus. To this end we also obtain an annealed CLT for the same system. We show that, almost surely, the variance of the quenched system is the same as for the annealed system. Our technique is the study of the transfer operator on an anisotropic Banach space specifically tailored to use the cone condition satisfied by the maps.
keywords: transfer operator. Central Limit Theorem iterated maps

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