Robust Exponential Decay of Correlations for Singular-Flows

Abstract

We construct open sets of C k (k ≥ 2) vector fields with singularities that have robust exponential decay of correlations with respect to the unique physical measure. In particular we prove that the geometric Lorenz attractor has exponential decay of correlations with respect to the unique physical measure.

Exponential Decay of Correlations for Piecewise Cone Hyperbolic Contact Flows

“…Since we may use the second bound of Lemma 4.1 instead of the first, this loss could occur only through the introduction of ∆ m in Step 1 of Lemma C.2 (see Lemma C.4), invoked in the third step of the present proof . In Step 1 of Lemma C.2, we got rid both of the x s and x u dependence of F (1) m . The x s dependence was a problem in Step 2 of Lemma C.2 (glueing).…”

“…The x s dependence was a problem in Step 2 of Lemma C.2 (glueing). If we give up this glueing step in Lemma C.2 then F (1) m can be allowed to depend on x s , to the cost of exponential growth (in Λ t , for Λ > 1 related to the dynamics). The x u dependence was a problem in Step 3 of Lemma C.2, where F (2) (Ax u , Bx s , x 0 ) could create exponential growth in the norm.…”

“…The x u dependence was a problem in Step 3 of Lemma C.2, where F (2) (Ax u , Bx s , x 0 ) could create exponential growth in the norm. Again, if we are willing to deal with an exponential factor, we can allow F (1) m to depend on x u . In other words, we can get rid of ∆ m in Lemma C.2.…”

“…It follows that the map L (1) (defined on a subset of R du ) also satisfies the inequalities defining D(C # ). In particular, this map is invertible, and we may define Γ (1) is defined on B(Πm, C 1/2 0 ). The map Ψ m .…”

“…The regularity of DF (1) m . To finish the proof, we should prove that DF (1) m satisfies the bounds defining K(C # ) for some constant C # independent of C 0 .…”