Ergodic Optimization for Hyperbolic Flows and Lorenz Attractors

We formulate abstract conditions under which a suspension flow satisfies the local central limit theorem. We check the validity of these conditions for several systems including reward renewal processes, Axiom A flows, as well as the systems admitting Young’s tower, such as Sinai’s billiard with finite horizon, suspensions over Pomeau–Manneville maps, and geometric Lorenz attractors.

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Convergence of moments for Axiom A and non-uniformly hyperbolic flows

Ergodic Theory and Dynamical Systems ◽

10.1017/s0143385711000174 ◽

2011 ◽

Vol 32 (3) ◽

pp. 1091-1100 ◽

Cited By ~ 13

Author(s):

IAN MELBOURNE ◽

ANDREI TÖRÖK

Keyword(s):

Hyperbolic Systems ◽

Finite Horizon ◽

Axiom A ◽

Hyperbolic Diffeomorphisms ◽

Convergence Of Moments ◽

Uniformly Hyperbolic Systems ◽

Hyperbolic Flows ◽

Lorenz Attractors ◽

Young Towers

AbstractIn the paper, we prove convergence of moments of all orders for Axiom A diffeomorphisms and flows. The same results hold for non-uniformly hyperbolic diffeomorphisms and flows modelled by Young towers with superpolynomial tails. For polynomial tails, we prove convergence of moments up to a certain order, and give examples where moments diverge when this order is exceeded. Non-uniformly hyperbolic systems covered by our result include Hénon-like attractors, Lorenz attractors, semidispersing billiards, finite horizon planar periodic Lorentz gases, and Pomeau–Manneville intermittency maps.

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Pressure Function and Limit Theorems for Almost Anosov Flows

Communications in Mathematical Physics ◽

10.1007/s00220-021-03962-x ◽

2021 ◽

Vol 382 (1) ◽

pp. 1-47

Author(s):

Henk Bruin ◽

Dalia Terhesiu ◽

Mike Todd

Keyword(s):

Thermodynamic Properties ◽

Central Limit ◽

Limit Theorems ◽

Central Limit Theorems ◽

Stable Laws ◽

Analytic Framework ◽

Pressure Function ◽

Hyperbolic Flows ◽

Anosov Flows ◽

Almost Anosov

AbstractWe obtain limit theorems (Stable Laws and Central Limit Theorems, both standard and non-standard) and thermodynamic properties for a class of non-uniformly hyperbolic flows: almost Anosov flows, constructed here. The link between the pressure function and limit theorems is studied in an abstract functional analytic framework, which may be applicable to other classes of non-uniformly hyperbolic flows.

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