A note on periodic orbits for singular-hyperbolic flows

We provide necessary and sufficient conditions for a suspension flow, over a subshift of finite type, to mix faster than any power of time. Then we show that these conditions are satisfied if the flow has two periodic orbits such that the ratio of the periods cannot be well approximated by rationals.

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Periodic Orbits for Hyperbolic Flows

American Journal of Mathematics ◽

10.2307/2373590 ◽

1972 ◽

Vol 94 (1) ◽

pp. 1 ◽

Cited By ~ 141

Author(s):

Rufus Bowen

Keyword(s):

Periodic Orbits ◽

Hyperbolic Flows

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Lalley's theorem on periodic orbits of hyperbolic flows

Ergodic Theory and Dynamical Systems ◽

10.1017/s0143385798100330 ◽

1998 ◽

Vol 18 (1) ◽

pp. 17-39 ◽

Cited By ~ 14

Author(s):

MARTINE BABILLOT ◽

FRANÇOIS LEDRAPPIER

Keyword(s):

Periodic Orbits ◽

Hyperbolic Flows

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Periodic orbits and holonomy for hyperbolic flows

Geometric and Probabilistic Structures in Dynamics - Contemporary Mathematics ◽

10.1090/conm/469/09172 ◽

2008 ◽

pp. 289-302 ◽

Cited By ~ 1

Author(s):

Mark Pollicott ◽

Richard Sharp

Keyword(s):

Periodic Orbits ◽

Hyperbolic Flows

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Livschitz Theorem in Suspension Flows and Markov Systems: Approach in Cohomology of Systems

Symmetry ◽

10.3390/sym12030338 ◽

2020 ◽

Vol 12 (3) ◽

pp. 338

Author(s):

Rosário D. Laureano

Keyword(s):

Periodic Orbits ◽

Systems Approach ◽

Existence Of Solutions ◽

Suspension Flows ◽

Markov Systems ◽

Fundamental Relationship ◽

Natural Notion ◽

Hyperbolic Flows ◽

Cohomological Equations

It is presented and proved a version of Livschitz Theorem for hyperbolic flows pragmatically oriented to the cohomological context. Previously, it is introduced the concept of cocycle and a natural notion of symmetry for cocycles. It is discussed the fundamental relationship between the existence of solutions of cohomological equations and the behavior of the cocycles along periodic orbits. The generalization of this theorem to a class of suspension flows is also discussed and proved. This generalization allows giving a different proof of the Livschitz Theorem for flows based on the construction of Markov systems for hyperbolic flows.

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