## Stability of mixing and rapid mixing for hyperbolic flows.(English)Zbl 1140.37004

This work focuses in $$C^r$$-stability of mixing and on the rate of mixing for Axiom A and Anosov flows.
Let $$M$$ be a compact connected manifold and let $$\Phi_t$$ be a $$C^1$$ flow on $$M$$. A $$\Phi_t$$-invariant set $$\Lambda$$ is (topologically) mixing if for any nonempty open sets $$U, V\subset\Lambda$$ there exists $$T>0$$ such that $$\Phi_t(U)\cap V\neq\emptyset$$ for all $$t> T$$. The flow is stably mixing if all nearby flows (in the appropriate topology) are mixing.
Let $$\mathcal{A}_r(M)$$ denote the set of $$C^r$$ flows ($$1\leq r\leq \infty$$) on $$M$$ satisfying the Axiom A and the no cycle property. It is well known that the nonwandering set $$\Omega$$ of such flows admits the spectral decomposition $$\Omega=\Lambda_1\cup\dots\cup\Lambda_k$$, where $$\Lambda_ i$$ are disjoint topologically transitive locally maximal hyperbolic sets. The sets $$\Lambda_i$$ are called (hyperbolic) basic sets. A basic set is nontrivial if it is neither an equlibrium nor a periodic solution.
One of main results in this work is the following
Theorem 1.1.
(a) Suppose $$2\leq r\leq\infty$$. There is a $$C^2$$-open, $$C^r$$-dense subset of flows in $$\mathcal{A}_r(M)$$ for which each nontrivial basic set is mixing.
(b) Suppose $$1\leq r\leq\infty$$. There is a $$C^1$$-open, $$C^r$$-dense subset of flows in $$\mathcal{A}_r(M)$$ for which each nontrivial attracting basic set is mixing.
The other main result (which extends the first one) shows that typical Axiom A flows are stably rapid mixing. Suppose that $$\Lambda$$ is a basic set for an Axiom A flow $$\Phi_t$$ and let $$\mu$$ be an equilibrium state for a Hölder potential. Given $$A, B\in L^2(\Lambda,\mu)$$, the correlation function is
$\rho_{A,B}(t)=\int_\Lambda A\circ\Phi_tB \,d\mu-\int_\Lambda A\,d\mu\int_\Lambda B\,d\mu$
The flow is mixing if and only if $$\rho_{A,B}(t) \to 0$$ as $$t\to\infty$$ for all $$A,B\in L^2(\Lambda,\mu)$$. The flow is rapid mixing if for any $$n>0$$, there is a constant $$C\geq 1$$ such that
$|\rho_{A,B}(t)|\leq C\|A\|\,\|B\|t^{-n}, \quad t>0$
for all observations $$A,B$$ that are sufficiently smooth in the flow direction. The notion of rapid mixing is independent of the choice of the equilibrium state $$\mu$$.
Theorem 1.6.
(a) Suppose $$2\leq r\leq\infty$$. There is a $$C^2$$-open, $$C^r$$-dense subset of flows in $$\mathcal{A}_r(M)$$ for which each nontrivial basic set is rapid mixing.
(b) Suppose $$1\leq r\leq\infty$$. There is a $$C^1$$-open, $$C^r$$-dense subset of flows in $$\mathcal{A}_r(M)$$ for which each nontrivial attracting basic set is rapid mixing.

### MSC:

 37A25 Ergodicity, mixing, rates of mixing 37D20 Uniformly hyperbolic systems (expanding, Anosov, Axiom A, etc.)

Axiom A flows
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