We consider the dynamical zeta functions of Selberg and Ruelle associated with the geodesic flow on a compact odd-dimensional hyperbolic manifold. These dynamical zeta functions are defined for a complex variable s in some right-half plane of $${\mathbb {C}}$$ C . In Spilioti (Ann Glob Anal Geom 53(2):151–203, 2018), it was proved that they admit a meromorphic continuation to the whole complex plane. In this paper, we establish functional equations for them, relating their values at s with those at $$-s$$ - s . We prove also a determinant representation of the zeta functions, using the regularized determinants of certain twisted differential operators.

中文翻译：

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非酉扭曲的 Selberg 和 Ruelle zeta 函数的函数方程

我们考虑与紧凑奇维双曲流形上的测地线流相关的 Selberg 和 Ruelle 的动态 zeta 函数。这些动态 zeta 函数是为 $${\mathbb {C}}$$ C 的某个右半平面中的复变量 s 定义的。在 Spilioti (Ann Glob Anal Geom 53(2):151–203, 2018) 中，证明他们承认对整个复平面的亚纯延续。在本文中，我们为它们建立了函数方程，将它们在 s 处的值与在 $$-s$$ - s 处的值相关联。我们还使用某些扭曲微分算子的正则化行列式证明了 zeta 函数的行列式表示。