Exponential decay of correlations for nonuniformly hyperbolic flows with a C^1+α stable foliation, including the classical Lorenz attractor

Annales Henri Poincaré. 17 (2016) 2975-3004

Vitor Araújo and Ian Melbourne

Abstract

We prove exponential decay of correlations for a class of C^1+α uniformly hyperbolic skew product flows, subject to a uniform nonintegrability condition. In particular, this establishes exponential decay of correlations for an open and dense set of geometric Lorenz attractors. As a special case, we show that the classical Lorenz attractor is robustly exponentially mixing.

Minor typos, etc (Last updated: February 2023)

In the statement of Lemma 2.23, |b| ≤ D' should be |b| ≥ D' (twice).

For the second statement of Proposition 2.18, it would be more natural to write:

∫_Y e^εr d Leb =∑_{h ∈ H} ∫_h(Y)e^εr d Leb =∑_{h ∈ H} ∫_Y e^{εr o h}|h'| d Leb ≤ ∑_{h ∈ H} e^{ε|r o h|_∞}|h'|_∞ which is finite by condition (iv). This proves the second statement since dμ / d Leb is bounded.

Condition (2.6) can be deleted.

Condition (2.5) would be more accurately written as ||h'x|-|h'y|| ≤ C₁|h'|_∞ |x-y|^α for all h ∈ H, x,y ∈ Y (and the preceding sentence could be erased). Observe that 1-t ≤ - log t for all t>0 and so a-b=a(1-b/a) ≤ a(log a-log b) for all a>b>0. Hence ||h'x|-|h'y|| ≤ |h'|_∞(log |h'x|-log |h'y|). Now apply condition (ii).

Similarly, condition (2.7) should be replaced by ||h'x|-|h'y|| ≤ C₂|h'|_∞ |x-y|^α ≤ C₂e^C₂|h'z| |x-y|^α for all h ∈ H_n, x,y,z ∈ Y