Manifolds and Lie Groups pp 283-299 | Cite as

# Some Stabilities of Group Automorphisms

Chapter

## Abstract

Let φ:M→M be a homeomorphism of a metric space (M,d) with distance function d. A (double) sequence {x_{i}}_{i∈M} of points x_{i} ∈ M is called a δ-pseudo-orbit of φ iff d(φ(x_{i}),x_{i+1}) ≤ δ for every i∈Z, where δ>0 is a constant (cf. [2]). Given ε>0, a δ-pseudo-orbit {x_{i}} is called to be ε-traced by a point y∈M iff d(φ^{i} (y),x_{i})≤ε for every i≤Z. We shall call φ stochastically stable, iff for any ε>0 there exists δ>0 such that every δ-pseudo-orbit of φ can be ε-traced by some point y∈M.

## Keywords

Riemannian Manifold Group Automorphism Stochastic Stability Linear Automorphism Jordan Canonical Form
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## Copyright information

© Springer Science+Business Media New York 1981