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Topological classification of linear hyperbolic cocycles

Abstract

In this paper linear hyperbolic cocycles are classified by the relation of topological conjugacy. Roughly speaking, two linear cocycles are conjugate if there exists a homeomorphism which maps their trajectories into each other. The problem of classification of discrete-time deterministic hyperbolic dynamical systems was investigated by Robbin (1972). He proved that there exist 4d classes ofd-dimensional deterministic discrete hyperbolic dynamical systems. We obtain a criterion for topological conjugacy of two linear hyperbolic cocycles and show that the number of classes depends crucially on the ergodic properties of the metric dynamical system over which they are defined. Our result is a generalization of the deterministic theorem of Robbin.

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Cong, N.D. Topological classification of linear hyperbolic cocycles. J Dyn Diff Equat 8, 427–467 (1996). https://doi.org/10.1007/BF02218762

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Key words

  • Linear hyperbolic cocycle
  • random homeomorphism
  • Oseledets splitting
  • orientation
  • dimension
  • coboundary

AMS Subject Classification

  • S8F15
  • 28D05
  • 58F19