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Hyperbolic Sets

  • D. V. Anosov
  • V. V. Solodov
Chapter
Part of the Encyclopaedia of Mathematical Sciences book series (EMS, volume 66)

Abstract

Hyperbolicity of a compact (two-sided) invariant set A (that is, a set consisting of entire trajectories) of a flow or cascade {g t } (t ∈ ℝ or t ∈ ℤ) given on a phase manifold M is defined in terms of the restriction over A of the tangent linear extension {Tg t } (see Anosov et al. 1985, Chap. 1, Sect. 2.2), that is, the properties of the DS {Tg t p −1 A}, where p : TMM is the natural projection. In other words, we are dealing with the behaviour of the solutions of the variational equations along the trajectories of our DS. (Speaking somewhat freely, we also have in mind not only “true” variational equations (and so on) for a flow, but also their analogues for a cascade {g k }. In the latter case, the role of the system of variational equations along the trajectory {g k x} is played by the representation of Tg k (x) in the form of the composite % MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamivaiaadE % gacaGGOaGaam4zamaaCaaaleqabaGaam4AaiabgkHiTiaaigdaaaGc % caWG4bGaaiykaiaad+gacqWIMaYscaWGVbGaamivaiaadEgacaGGOa % Gaam4zaiaadIhacaGGPaGaam4BaiaadsfacaWGNbGaaiikaiaadIha % caGGPaGaaiilaaaa!4B9A!\[Tg({g^{k - 1}}x)o \ldots oTg(gx)oTg(x),\] while the role of the solutions of this system (which is more important) is played by the function of discrete time kTg k (x)ς with ς ∈ TxM. The map Tg k (x) plays the role of the Cauchy matrix of the system of variational equations along the trajectory {g k x}.) It is worth recalling, in connection with the term “tangent linear extension”, that in (Anosov et al. 1985, Chap. 3, Sect. 5.2) the more general notion of a linear extension of a DS was introduced. In this terminology the tangent linear extension {Tg t } is, in fact, a linear extension of the DS {g t }, while its restriction to the closed invariant subset TMA = p −1 A, which is a vector bundle over A, is the linear extension of the DS {g t A}.2

Keywords

Periodic Point Unstable Manifold Linear Extension Topological Entropy Periodic Trajectory 
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Authors and Affiliations

  • D. V. Anosov
  • V. V. Solodov

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