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RUELLE TRANSFER OPERATORS WITH TWO COMPLEX PARAMETERS AND APPLICATIONS

机译:具有两个复杂参数的RUELLE转移算子和应用

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For a C~2 Axiom A flow ø_t : M → M on a Riemannian manifold M and a basic set A for ø_t we consider the Ruelle transfer operator L_(f-Sτ+zg), where f and g are real-valued Holder functions on Λ, τ is the roof function and s,z ∈ C are complex parameters. Under some assumptions about ø_t we establish estimates for the iterations of this Ruelle operator in the spirit of the estimates for operators with one complex parameter (see [4], [21], [22]). Two cases are covered: (i) for arbitrary Hölder f, g when |Imz| ≤ B|Ims|~µ for some constants B > 0, 0 < µ < 1 (µ = 1 for Lipschitz f,g), (ii) for Lipschitz f, g when |Ims| ≤ B_1|Imz| for some constant B_1 > 0 . Applying these estimates, we obtain a non zero analytic extension of the zeta function ζ(s, z) for P_f - ∈ < Re(s) < P_f and |z| small enough with a simple pole at s = s(z). Two other applications are considered as well: the first concerns the Hannay-Ozorio de Almeida sum formula, while the second deals with the asymptotic of the counting function π_F(T) for weighted primitive periods of the flow ø_t.
机译:对于C〜2公理A的黎曼流形M上的流体ø_t:M→M和ø_t的基本集合A,我们考虑Ruelle转移算子L_(f-Sτ+ zg),其中f和g是实值Holder函数在Λ上,τ是屋顶函数,而s,z∈C是复参数。在关于ø_t的一些假设下,我们本着针对具有一个复杂参数的算子的估计精神,为该Ruelle算子的迭代建立了估计(请参见[4],[21],[22])。涉及两种情况:(i)当| Imz |时对于任意Hölderf,g对于某些常数B> 0,0 <µ <1(对于Lipschitz f,g,µ = 1),(ii)对于Lipschitz f,g,当|| Ims |时,≤B | Ims |〜µ ≤B_1 | Imz |对于一些常数B_1> 0。应用这些估计,我们获得了P_f-∈

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