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**On contact Anosov flows.**
*(English)*
Zbl 1067.37031

In this beautiful paper the author proves exponential decay of correlations for a contact Anosov flow \(T_t\) of class \(C^4\) on a closed manifold \(M\). The main idea is to establish contraction for the action of the flow on the space \(C^1 (M, \mathbb C)\) of continuously differentiable functions on \(M\) by comparing different norms on \(C^1 (M, \mathbb C)\). By the Anosov property, for sufficient small \(\lambda > 0\) the pseudo-distance \(d_s\) on \(M\) defined by \(d_s (x, y) = \int^\infty _0 \, e^{\lambda t} d (T_t x , T_t y) \,dt\) restricts to a finite distance function on each strong stable manifold for \(T_t\). For \(\beta \in [0, 1]\) and sufficiently small \(\delta > 0\), one obtains a norm \(| \, |_{s, \beta}\) on \(C^1 (M, \mathbb C)\) by defining \(| \varphi|_{s, \beta} = | \varphi |_\infty + \sup _{d_s (x, y) \leq \delta} \frac{| \varphi (x) - \varphi (y) |}{d_s (x, y)^\beta}\). The author then shows that for sufficiently small \(\beta > 0\), there is a constant \(\sigma >0\) such that \(\int \, f (\varphi \circ T_t) = \int f \int \varphi + O (e^{-\sigma t} | f |_{C^1} | \varphi |_{s, \beta})\) for all \(f, \varphi \in C^1 (M, \mathbb C)\) which proves exponential decay of correlations. Interestingly, the spectral gap property for the operators \(L_t\, f = f \circ T_t \, (t > 0)\) is not established.

The estimate follows from the main result of the paper which can be formulated as follows: For small \(\beta \in [0,1]\) denote by \(\mathcal D_ \beta\) the unit ball in the completion of \(C^1 (M, \mathbb C)\) with respect to the norm \(|\, |_{s, \beta}\). For small \(\lambda > 0\) define \(d_u (x, y) = \int^0 _{-\infty} e^{- \lambda t} d (T_t x, T_t y)dt\) and let \(\| f \| = \sup_{d_u (x, y) \leq \delta} \frac{| f (x) - f [y) |}{d_u (x, y)^ \beta} + | f |_\infty\). Then the operators \(L_t (t > 0)\) form a strongly continuous group on the completion of \(C^1 (M, \mathbb C)\) with respect to the norm \(\| \;\|\) whose spectral radius is bounded by one. Moreover, there are numbers \(\sigma > 0\) and \(c > 0\) such that for every function \(f\) of class \(C^1\) with \(\int f = 0\), we have \(\| L_t f \| \leq c\, e^{-\sigma t} | f |_{C^1}\).

The proof of this result consists in a careful analysis of the resolvent of the generator of the flow acting on various function spaces. It avoids heavy machinery and is short and entirely selfcontained. Its technically most difficult part is an estimate which is motivated by an earlier result of D. Dolgopyat [Ann. Math. (2) 147, 357–390 (1998; Zbl 0911.58029)]. The paper is very well written and summarizes background material known to the experts in three appendices.

The estimate follows from the main result of the paper which can be formulated as follows: For small \(\beta \in [0,1]\) denote by \(\mathcal D_ \beta\) the unit ball in the completion of \(C^1 (M, \mathbb C)\) with respect to the norm \(|\, |_{s, \beta}\). For small \(\lambda > 0\) define \(d_u (x, y) = \int^0 _{-\infty} e^{- \lambda t} d (T_t x, T_t y)dt\) and let \(\| f \| = \sup_{d_u (x, y) \leq \delta} \frac{| f (x) - f [y) |}{d_u (x, y)^ \beta} + | f |_\infty\). Then the operators \(L_t (t > 0)\) form a strongly continuous group on the completion of \(C^1 (M, \mathbb C)\) with respect to the norm \(\| \;\|\) whose spectral radius is bounded by one. Moreover, there are numbers \(\sigma > 0\) and \(c > 0\) such that for every function \(f\) of class \(C^1\) with \(\int f = 0\), we have \(\| L_t f \| \leq c\, e^{-\sigma t} | f |_{C^1}\).

The proof of this result consists in a careful analysis of the resolvent of the generator of the flow acting on various function spaces. It avoids heavy machinery and is short and entirely selfcontained. Its technically most difficult part is an estimate which is motivated by an earlier result of D. Dolgopyat [Ann. Math. (2) 147, 357–390 (1998; Zbl 0911.58029)]. The paper is very well written and summarizes background material known to the experts in three appendices.

Reviewer: Ursula Hamenstädt (Bonn)

### MSC:

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

37A25 | Ergodicity, mixing, rates of mixing |