Small random perturbations of dynamical systems and the definition of attractors
RESEAUX
SYSTEMES DYNAMIQUES
PHYSIQUE STATISTIQUE
SYSTEMES COMPLEXES
THEORIES NON LINEAIRES
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.
RUELLE
P/81/23
IHES
03/1981
A4
15 f.
EN
TEXTE
PREPUBLICATION
P_81_23.pdf
1981
A Heuristic theory of phase transitions
RESEAUX
SYSTEMES COMPLEXES
ESPACES DE BANACH
TRANSITIONS DE PHASE
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.
RUELLE
P/76/149
IHES
10/1976
A4
25 f.
EN
TEXTE
PREPUBLICATION
P_76_149.pdf
1976
Some Remarks on the location of zeroes of the partition function for lattice systems
RESEAUX
SYSTEMES COMPLEXES
SYSTEMES DYNAMIQUES
SYSTEMES NON LINEAIRES
THEORIE DES TREILLLIS
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.
RUELLE
P/72/29
IHES
1972
A4
23 f.
EN
TEXTE
PREPUBLICATION
P_72_29.pdf
1972
Bifurcations in the presence of a symmetry group
RESEAUX
SYSTEMES COMPLEXES
SYSTEMES DYNAMIQUES
SYSTEMES NON LINEAIRES
GROUPES DE SYMETRIE
ELECTROMAGNETISME
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.
RUELLE
P/72/20
IHES
07/1972
A4
33 f.
EN
TEXTE
PREPUBLICATION
P_72_20.pdf
1972
On the Nature of turbulence
RESEAUX
PHYSIQUE STATISTIQUE
SYSTEMES COMPLEXES
SYSTEMES DYNAMIQUES
INFORMATIQUE QUANTIQUE
SYSTEMES NON LINEAIRES
Abstract : A mechanism for the genetration of turbulence and related phenomena in dissipative systems is proposed.
RUELLE
TAKENS
P/70/X036
IHES
05/1970
A4
16 f.
EN
TEXTE
PREPUBLICATION
P_70_X036.pdf
1970
Integral representation of states on a C*-algebras
RESEAUX
PHYSIQUE STATISTIQUE
SYSTEMES COMPLEXES
SYSTEMES DYNAMIQUES
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.
RUELLE
P/70/X029
IHES
1970
A4
30 f.
EN
TEXTE
PREPUBLICATION
P_70_X029.pdf
1970
A Variational formulation of equilibrium statistical mechanics and the Gibbs phase rule
ENTROPIE
RESEAUX
PHYSIQUE STATISTIQUE
SYSTEMES COMPLEXES
Abstract : It is shown that for an infinite lattice system, thermodynamic equilibrium is the solution of a variational problem involving a mean entropy of states introduced earlier [2]. As an application, a version of the Gibbs phase rule is proved.
RUELLE
P/67/X012
IHES
1967
A4
6 f.
EN
TEXTE
PREPUBLICATION
P_67_X012.pdf
1967
Condensation of lattice gases
RESEAUX
PHYSIQUE STATISTIQUE
SYSTEMES COMPLEXES
CONDENSATION
GAZ
RUELLE
GINIBRE
GROSSMANN
P/66/X005
IHES
1966
A4
8 f.
EN
TEXTE
PREPUBLICATION
P_66_X005.pdf
1966
Mean entropy of States in classical statistical mechanics
ENTROPIE
RESEAUX
PHYSIQUE STATISTIQUE
SYSTEMES COMPLEXES
Abstract : The equilibrium states for an infinite system of classical mechanics may be represented by states over Abelian C* algebras. We consider here continuous and lattice systems and define a mean entropy for their states. The properties of this mean entropy are investigated : linearity, upper semi-continuity, integral representations. In the lattice case, it is found that our mean entropy coincides with the KOLMOGOROV-SINAI invariant of ergodic theory.
RUELLE
ROBINSON
P/66/04
IHES
1966
A4
25 f.
EN
TEXTE
PREPUBLICATION
P_66_04.pdf
1966
On the Existence of fixed points of the composition operator for circle maps
RESEAUX CEREBRAUX
PHYSIQUE STATISTIQUE
SYSTEMES COMPLEXES
THEORIES NON LINEAIRES
DYNAMIQUE
INFORMATIQUE QUANTIQUE
Abstract : In the theory of circle maps with golden ratio rotation number formulated by Feigenbaum, Kadanoff, and Shenker [FKS], and by Ostlund, Rand, Sethna, and Siggia [ORSS], a central role is played by fixed points of a certain composition operator in map space. We define a common setting for the problem of proving the existence of these fixed points and of those occurring in the theory of maps of the interval. We give a proof of the existence of the fixed points for a wide range of the parameters on which they depend.
EPSTEIN
ECKMANN
P/86/29
IHES
05/1986
A4
13 f.
EN
TEXTE
PREPUBLICATION
P_86_29.pdf
1986
Scaling of Mandelbrot sets generated by critical point preperiodicity
RESEAUX CEREBRAUX
PHYSIQUE STATISTIQUE
SYSTEMES COMPLEXES
THEORIES NON LINEAIRES
DYNAMIQUE
Astract : Letz?f?(z) be a complex holomorphic function depending holomorphically on the complex parameter ?. If, for ?=0, a critical point off0 falls after a finite number of steps onto an unstable fixed point off0, then, in the parameter space, near 0, an infinity of more and more accurate copies of the Mandelbrot set appears. We compute their scaling properties.
EPSTEIN
ECKMANN
P/83/70
IHES
11/1983
A4
8 f.
EN
TEXTE
PREPUBLICATION
P_83_70.pdf
1983
Analyticity properties of the Feigenbaum function
RESEAUX CEREBRAUX
PHYSIQUE STATISTIQUE
SYSTEMES COMPLEXES
THEORIES NON LINEAIRES
DYNAMIQUE
INFORMATIQUE QUANTIQUE
Absract : Analyticity properties of the Feigenbaum function [a solution ofg(x)=???1g(g(?x)) withg(0)=1,g?(0)=0,g?(0)<0] are investigated by studying its inverse function which turns out to be Herglotz or anti-Herglotz on all its sheets. It is found thatg is analytic and uniform in a domain with a natural boundary.
EPSTEIN
LASCOUX
P/81/27
IHES
05/1981
A4
27 f.
EN
TEXTE
PREPUBLICATION
P_81_27.pdf
1981
Time-ordered products and Schwinger functions
RESEAUX CEREBRAUX
THEORIE DES CHAMPS
SYSTEMES COMPLEXES
PHYSIQUE STATISTIQUE
Abstract : It is shown that every system of time-ordered products for a local field theory determines a related system of Schwinger functions possessing an extended form of Osterwalder-Schrader positivity and that the converse is true provided certain growth conditions are satisfied. This is applied to the ? 3 4 theory and it is shown that the time-ordered functions andS-matrix elements admit the standard perturbation series as asymptotic expansions.
EPSTEIN
ECKMANN
P/78/227
IHES
1978
A4
35 f.
EN
TEXTE
PREPUBLICATION
P_78_227.pdf
1978
Remarks on two theorems of E. Lieb
RESEAUX CEREBRAUX
PHYSIQUE STATISTIQUE
SYSTEMES COMPLEXES
DYNAMIQUE
THEORIES NON LINEAIRES
INFORMATIQUE QUANTIQUE
Abstract : The concavity of two functions of a positive matrixA, Tr exp(B + logA) and TrA r KA p K* (whereB=B* andK are fixed matrices), recently proved by Lieb, can also be obtained by using the theory of Herglotz functions.
EPSTEIN
P/73/41
IHES
02/1973
A4
9 f.
EN
TEXTE
PREPUBLICATION
P_73_41.pdf
1973
Renormalization of non polynomial Lagrangians in Jaffe's class
RESEAUX CEREBRAUX
PHYSIQUE STATISTIQUE
SYSTEMES COMPLEXES
DYNAMIQUE
THEORIES NON LINEAIRES
INFORMATIQUE QUANTIQUE
Abstract : t, It is shown how a renormalized perturbation series can be defined for a
theory with strictly locaI, non-polynomial, interacting Lagrangian
:A(x)r: 2e(x) = )__, t,-----
r=O r!
so as to preserve locality at every order.
EPSTEIN
GLASER
P/72/10
IHES
1972
A4
11 f.
EN
TEXTE
PREPUBLICATION
P_72_10.pdf
1972