The Density at infinity of a discrete group of hyperbolic motions
On the Ergodic theory at infinity of an arbitrary discrete group of hyperbolic motions
A Fioliation of geodesics in characterized by having no tangent homologies and a homological characterization of foliations consisting of minimal surfaces
Dedicated to the memory of George Cooke.
Hyperbolic geometry and homomorphisms. (Suivi de) On Complexes that are Lipschitz manifolds
Two papers for the proceedings of the Athnes, Georgia Topology Conference of August, 1977
Cycles for the dynamical study of foliated manifolds and complex manifolds
A Counterexample to the periodic orbit conjecture
The Homology theory of the closed geodesic problem
A semi-local combinatorial formula for the signature of a 4k-manifold
Deligne P. / Griffiths, P. / Morgan, J. et al. Real homotopy theory of Kähler manifolds. Invent Math 29 p. 245–274 (1975). https://doi.org/10.1007/BF01389853
Abstract : We discuss the possibility that, besides periodic and quasiperiodic crystals, there exist turbulent crystals as thermodynamic equilibrium states at non-zero temperature. Turbulent crystals would not be invariant under translation, but would differ from other crystals by the fuzziness of some diffraction peaks. Turbulent crystals could appear by breakdown of long range order in quasiperiodic crystals with two independent modulations.
Small random perturbations of dynamical systems and the definition of attractors
Abstract : The strange attractors plotted by computers and seen in physical experiments do not necessarily have an open basin of attraction. In view of this we study a new definition of attractors based on ideas of Conley. We argue that the attractors observed in the presence of small random perturbations correspond to this new definition.
Characteristic exponents and invariant manifolds in Hilbert space
Abstract : The multiplicative ergodic theorem and the construction almost everywhere of stable and unstable manifolds (Pesin theory) are extended to differentiable dynamical systems on Hilbert manifolds under some compactness assumptions. The results apply to partial differential equations of evolution and also to non-invertible maps of compact manifolds.
Abstract :Recent attempts at understanding hydrodynamic turbulence have used the ideas of strange attractors, characteristic exponents and stable manifolds for differentiable dynamical systems in finite dimensional spaces. This use was somewhat metophorical, because hydrodynamic evolution is defined in infinite dimensional functional spaces. A recent study indicates that many results in infinite dimensional Hilbert spaces under certain compactness assumptions. This is the case in particular for the time evolution defined by the Navier-Stokes equations in a bounded region of R2 or R3.
On the Measures which describe turbulence
Abstract : One expects that the average behavior (over large times) for hydordynamics and other natural phenomena is described by certain asymptotic measures on phase space. If initial conditions in a set of zero Lebesgu measure ar discarded, the asymptotic measures can be characterized on the basis of heuristic arguments. The requirement of stability under small stochastic perturbations produces measures with the same characterizations. We give here a critical discussion of the heuristic arguments and of the possible use of the characterizations of the asymptotic measures in the study of turbulence.
Ergodic theory of differentiable dynamical systems
Abstract : If f is a C1+? diffeomorphism of a compact manifold M, we prove the existence of stable manifolds, almost verywhere with respect to every f-invariant probability measure on M. These stable manifolds are smooth but do not in general constitute a continuous family. The proof of this stable manifold theorem (and similar results) is through the study of random matrix products (multiplicative ergodic theorem) and perturbation of such products.
Analiticity properties of the characteristic exponents of random matrix products
Sensitive dependence on initial conditions and turbulent behavior of dynamical systems
Abstract : The asymptotic behavior of differentiable dynamical systems is analyzed. We discuss its descriptoin by asymptotic measures and the turbulent behavior with senditive dependence on initial condition.
Dynamical systems with turbulent behavior
[Text of a talk presented at the International Mathematical Physics Conference in Rome, 1977]
Integral representation of measures associated with a foliation
Applications conservant une mesure absolument continue par rapport à dx sur [0, 1]
Abstract : Sufficient conditions are given such that a differentiable, noninvertible, map g : [0,1]~[0,1] leaves invariant a measure absolutely continuous with respect to the Lebesgue measure. In particular, this is shown to be the case for 9(x)=Rx(1- x) when R = 3,6785735 ....
Abstract : Let Z be a suitable Banach space of interactions for a lattice spin system. If n+1 thermodynamic phases coexist for ?0 ?Z, it is shown that a manifold of codimension n of coexistence of (at least) n+1 phases passes through ?0. There are also n+1 manifolds of codimension n?1 of coexistence of (at least) n phases; these have a common boundary along the manifold of coexistence of n+1 phases. And so on for coexistence of fewer phases. This theorem is proved under a technical condition (R) which says that the pressure is a differentiable function of the interaction at ?0 when restricted to some codimensionn affine subspace of Z. The condition (R) has not been checked in any specific instance, and it is possible that our theorem is useless or vacuous. We believe however that the method of proof is physically correct and constitutes at least a heuristic proof of the Gibbs phase rule.
Probability estimates for continuous spin systems
Abstract : Probability estimates for classical systems of particles with superstable interactions [1] are extended to continuous spin systems.
Abstract : Using a theorem on convex functions due to Israel, it is shown that a point of coexistence of n+1n+1 phases cannot be isolated in the space of interactions, but lies on some infinite dimensional manifold.
Zeta-functions for expanding maps and Anosov flows
Abstract : Given a real-analytic expanding endomorphism of a compact manifold M, a meromorphic zeta function is defined on the complex-valued real-analytic functions on M. A zeta function for Anosov flows is shown to be meromorphic if the flow and its stable-unstable foliations are real-analytic.
Generalized zeta-functions for axiom : a basic sets
Some Remarks on the location of zeroes of the partition function for lattice systems
Abstract : We use techniques which generalize the Lee-Yang circle theorem to investigate the distribution of zeroes of the partition function for various classes of classical lattice systems.
Bifurcations in the presence of a symmetry group
Abstract : Let the origin O of a Banach space E be a fixed point of a diffeomorphism or a critical point of a vector, assuming equivariance under a linear group of isometries of E. Explicit techniques are presented to handle the generalization of the Hopf bifurcation to this equivariant situation.
Statistical mechanics on a compact set withe Z? action satisfying expansiveness and specification
Abstract : A mechanism for the genetration of turbulence and related phenomena in dissipative systems is proposed.
Analiticity of Green's functions of dilute quantum gases
Superstable interactions in classical statistical mechanics
Abstract : We consider classical systems of particles inv dimensions. For a very large class of pair potentials (superstable lower regular potentials) it is shown that the correlation functions have bounds of the form ?(x1,...,xn)??n. Using these and further inequalities one can extend various results obtained by Dobrushin and Minlos [3] for the case of potentials which are non-integrably divergent at the origin. In particular it is shown that the pressure is a continuous function of the density. Infinite system equilibrium states are also defined and studied by analogy with the work of Dobrushin [2a] and of Lanford and Ruelle [11] for lattice gases.
Integral representation of states on a C*-algebras
Abstract : Let E be the compact set of states on a C?-algebra U with identity. We discuss the representations of a state ? as barycenter of a probability measure ? on E. Examples of such representations are the central decomposition and the ergodic decomposition. They are associated with an Abelian von Neumann algebra B in the commutant ?(U)? of the image of U in the representation canonically associated with ?. This situation is studied in general and a number of applications are discussed.