1. The Lorenz attractor is mixing
Stefano Luzzatto, Imperial College London
(with Ian Melbourne and Frederic Paccaut)

2. The Lorenz Equations x’ = 10(y-x) y’= 28x-y-xz z’= xy-(8/3)z
Motivation:
Geophysics & Meterology - Lorenz ‘63
Mathematics - Ruelle & Takens ‘71

3. Afraimovich,Bykov,Silnikov,Guckenheimer,Williams’77’79
Theorem: The Lorenz attractor has a unique ergodic SRB measure with a positive Lyapunov exponent.
Afraimovich,Bykov,Silnikov,Guckenheimer,Williams’77’79
Geometric Model of the Lorenz attractor
Robinson ‘84, Pesin ‘92
SRB measures for geometric model
Tucker - ‘02
Rigorous Numerics: results for geometric model apply to the original equations
Lots of other work on Lorenz equations/model.
Ratner - Mixing implies Bernoulli - ‘78
Kifer - Random perturbations - ‘88

4. Require some “non-resonance” in the flow return times.
Mixing: every sets spreads itself out evenly under the flow according to the invariant measure.
Flow direction is not expanding and does not contribute to mixing. Constant time suspension flows are never mixing.
Require some “non-resonance” in the flow return times.

5. Theorem: Lorenz flows are stably mixing
Luzzatto-Melbourne-Paccaut. ‘04.
Contact Anosov flows are stably mixing
Katok and Burns ‘94 (inc Geodesic flows - Anosov ‘68)
Stable mixing is open and dense in class of
codimension 1 Anosov flows - Plante ‘72
Anosov flows which have a global cross-section which is an infra-nil manifold - Bowen ‘76
All Axiom A flows - Field-Melbourne-Torok ‘03.

6. Idea of proof
Tucker’s hyperbolicity estimates imply the existence of a global invariant C1+epsilon stable foliation as in the geometric model.

7. Lorenz-like semi-flows
Base transformation: f: I -> I
C1+epsilon
Uniformly expanding
Infinite derivative at discontinuity
Locally eventually onto
Roof function
f(x) ~ log x

8. Intuitively equilibrium point aids mixing: nearby points are separated as they pass close to the discontinuity.

9. Key ingredients The (non-Markov) one-dimensional map f
admits an induced Markov structure
(Diaz-Ordaz ‘04)
Livsic regularity for maps which admit
induced Markov structures
(Bruin-Holland-Nicol ‘04)

10. These conditions imply that mixing can
be obtained by (not) solving a
cohomological equation:
There do NOT exist
continuous such that

11. Extensions and generalizations
Arguments essentially works for suspension flows with unbounded roof functions over nonuniformly expanding or nonuniformly hyperbolic maps.
Lorenz attractors without global stable foliations (Pesin)
Contracting Lorenz attractor (Rovella,Metsger)
Lorenz attractors with hooks (Luzzatto-Viana, Luzzatto-Holland)