##
**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.

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.

Reviewer: Carlos Vasquez (Valparaiso)

### MSC:

37A25 | Ergodicity, mixing, rates of mixing |

37D20 | Uniformly hyperbolic systems (expanding, Anosov, Axiom A, etc.) |