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Ergodic Theory: Rigidity

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Encyclopedia of Complexity and Systems Science

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Definition of the Subject

As one can see from this volume, chaotic behavior of complex dynamical systems is prevalent in nature and in large classes oftransformations. Rigidity theory can be viewed as the counterpart to the generic theory of dynamical systems which often investigates chaotic dynamics fora typical transformation belonging to a large class. In rigidity one is interested in finding obstructions to chaotic, or generic,behavior. This often leads to rather unexpected classification results. As such, rigidity in dynamics and ergodic theory is difficult to define preciselyand the best approach to this subject is to study various results and themes that developed so far. A classification is offered below in local,global, differentiable and measurable rigidity. One should note that all branches are strongly intertwined and, at this stage of the development of thesubject, it is difficult to separate them.

Rigidity is a well developed and prominent topic in modern mathematics....

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Abbreviations

Differentiable rigidity:

Differentiable rigidity refers to finding invariants to the differentiable conjugacy of dynamical systems, and, more general, group actions.

Local rigidity:

Local rigidity refers to the study of perturbations of homomorphisms from discrete or continuous groups into diffeomorphism groups.

Global rigidity:

Global rigidity refers to the classification of all group actions on manifolds satisfying certain conditions.

Measure rigidity:

Measure rigidity refers to the study of invariant measures for actions of abelian groups and semigroups.

Lattice:

A lattice in a Lie group is a discrete subgroup of finite covolume.

Conjugacy:

Two elements \( { g_1,g_2 } \) in a group G are said to be conjugated if there exists an element \( { h\in G } \) such that \( { g_1=h^{-1}g_2h } \). The element h is called conjugacy.

C k Conjugacy:

Two diffeomorphisms \( { \phi_1, \phi_2 } \) acting on the same manifold M are said to be C k-conjugated if there exists a C k diffeomorphism h of M such that \( { \phi_1=h^{-1}\circ \phi_2\circ h } \). The diffeomorphism h is called C k conjugacy.

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Acknowledgment

This research was supported in part by NSF Grant DMS-0500832.

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  1. West Chester University, West Chester, USA

    Viorel Niţică

  2. Institute of Mathematics, Bucharest, Romania

    Viorel Niţică

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    Robert A. Meyers Ph. D. (Editor-in-Chief) (Editor-in-Chief)

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Niţică, V. (2009). Ergodic Theory: Rigidity. In: Meyers, R. (eds) Encyclopedia of Complexity and Systems Science. Springer, New York, NY. https://doi.org/10.1007/978-0-387-30440-3_185

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