On the spectral and mixing properties of rank-one constructions in ergodic theory

In this paper we investigate the rigidity of symmetric spaces of non-compact type using ergodic theory such as Patterson–Sullivan measure and the marked length spectrum along with the cross ratio on the limit set. In particular, we prove that the marked length spectrum determines the Zariski dense subgroup up to conjugacy in the isometry group of the product of rank-one symmetric spaces. As an application, we show that two convex cocompact, negatively curved, locally symmetric manifolds are isometric if the Thurston distance is zero and the critical exponents of the Poincaré series are the same, and the same is true if the geodesic stretch is equal to one.

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GALILEI-INVARIANT QUANTUM FIELD THEORIES WITH PAIR INTERACTIONS: A REVIEW

International Journal of Modern Physics A ◽

10.1142/s0217751x91001453 ◽

1991 ◽

Vol 06 (17) ◽

pp. 2937-2970 ◽

Cited By ~ 12

Author(s):

HEiDE NARNHOFER ◽

WALTER THIRRING

Keyword(s):

Ergodic Theory ◽

Field Theories ◽

Quantum Field Theories ◽

Quantum Field ◽

Mixing Properties ◽

Essential Condition ◽

Classical Systems ◽

Pair Potentials ◽

Heisenberg Representation ◽

Invariant Quantum

We exhibit a class of quantum field theories where particles interact with pair potentials and for which the time evolution exists in the Heisenberg representation. The essential condition for existence is stability in the thermodynamic sense and this is achieved by having the interaction fall off with the relative momenta of the particles. This can be done in a Galilei-invariant manner. We show that these systems have some mixing properties which one postulates in ergodic theory but which are difficult to prove for classical systems.

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Poisson suspensions and infinite ergodic theory

Ergodic Theory and Dynamical Systems ◽

10.1017/s0143385708080279 ◽

2009 ◽

Vol 29 (2) ◽

pp. 667-683 ◽

Cited By ~ 16

Author(s):

EMMANUEL ROY

Keyword(s):

Dynamical Systems ◽

Spectral Theory ◽

Ergodic Theory ◽

Mixing Properties ◽

New Approach ◽

Infinite Measure ◽

Infinite Ergodic Theory

AbstractWe investigate the ergodic theory of Poisson suspensions. In the process, we establish close connections between finite and infinite measure-preserving ergodic theory. Poisson suspensions thus provide a new approach to infinite-measure ergodic theory. Fields investigated here are mixing properties, spectral theory, joinings. We also compare Poisson suspensions to the apparently similar looking Gaussian dynamical systems.

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