109 research outputs found
Fredholm determinants, Anosov maps and Ruelle resonances
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 \Co^\infty Anosov diffeomorphisms, the dynamical
determinant is an entire function
On Contact Anosov Flows
Exponential decay of correlations for \Co^{(4)} Contact Anosov flows is
established. This implies, in particular, exponential decay of correlations for
all smooth geodesic flows in strictly negative curvature
Banach spaces adapted to Anosov systems
We study the spectral properties of the Ruelle-Perron-Frobenius operator
associated to an Anosov map on classes of functions with high smoothness. To
this end we construct anisotropic Banach spaces of distributions on which the
transfer operator has a small essential spectrum. In the C^\infty case, the
essential spectral radius is arbitrarily small, which yields a description of
the correlations with arbitrary precision. Moreover, we obtain sharp spectral
stability results for deterministic and random perturbations. In particular, we
obtain differentiability results for spectral data (which imply
differentiability of the SRB measure, the variance for the CLT, the rates of
decay for smooth observable, etc.).Comment: 26 page
Map Lattices coupled by collisions
We introduce a new coupled map lattice model in which the weak interaction
takes place via rare "collisions". By "collision" we mean a strong (possibly
discontinuous) change in the system. For such models we prove uniqueness of the
SRB measure and exponential space-time decay of correlations
Toward the Fourier law for a weakly interacting anharmonic crystal
For a system of weakly interacting anharmonic oscillators, perturbed by an
energy preserving stochastic dynamics, we prove an autonomous (stochastic)
evolution for the energies at large time scale (with respect to the coupling
parameter). It turn out that this macroscopic evolution is given by the so
called conservative (non-gradient) Ginzburg-Landau system of stochastic
differential equations. The proof exploits hypocoercivity and hypoellipticity
properties of the uncoupled dynamics
Convergence to equilibrium for intermittent symplectic maps
We investigate a class of area preserving non-uniformly hyperbolic maps of
the two torus. First we establish some results on the regularity of the
invariant foliations, then we use this knowledge to estimate the rate of
mixing.Comment: LaTeX, 23 page
Limit Theorems for Fast-slow partially hyperbolic systems
We prove several limit theorems for a simple class of partially hyperbolic
fast-slow systems. We start with some well know results on averaging, then we
give a substantial refinement of known large (and moderate) deviation results
and conclude with a completely new result (a local limit theorem) on the
distribution of the process determined by the fluctuations around the average.
The method of proof is based on a mixture of standard pairs and Transfer
Operators that we expect to be applicable in a much wider generality
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