On the number of poles of the dynamical zeta functions for billiard flow
Copyright policy )On the number of poles of the dynamical zeta functions for billiard flow
We study the number of the poles of the meromorphic continuation of the dynamical zeta functions $η_N$ and $η_D$ for several strictly convex disjoint obstacles satisfying non-eclipse condition. We obtain a strip $\{z \in \mathbb C:\: {\rm Re}\: s > β\}$ with infinite number of poles. For $η_D$ we prove the same result assuming the boundary real analytic. Moreover, for $η_N$ we obtain a characterisation of $β$ by the pressure $P(2G)$ of some function $G$ on the space $Σ_A^f$ related to the dynamical characteristics of the obstacle.
This is revised version including shorter proofs of some results. The paper is accepted for publication in Discrete and Continuous Dynamical Systems
Uniformly hyperbolic systems (expanding, Anosov, Axiom A, etc.), Selberg zeta functions and regularized determinants; applications to spectral theory, Dirichlet series, Eisenstein series, etc. (explicit formulas), dynamical zeta function, local trace formula, Functional analytic techniques in dynamical systems; zeta functions, (Ruelle-Frobenius) transfer operators, etc., Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.), 37D20, 37D05, 37D40, 11M41, 11M36, billiard flow, Dynamical Systems (math.DS), strip with infinite number of poles, Dynamical systems with hyperbolic orbits and sets, Dynamical systems with singularities (billiards, etc.), FOS: Mathematics, Mathematics - Dynamical Systems, Other Dirichlet series and zeta functions
Uniformly hyperbolic systems (expanding, Anosov, Axiom A, etc.), Selberg zeta functions and regularized determinants; applications to spectral theory, Dirichlet series, Eisenstein series, etc. (explicit formulas), dynamical zeta function, local trace formula, Functional analytic techniques in dynamical systems; zeta functions, (Ruelle-Frobenius) transfer operators, etc., Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.), 37D20, 37D05, 37D40, 11M41, 11M36, billiard flow, Dynamical Systems (math.DS), strip with infinite number of poles, Dynamical systems with hyperbolic orbits and sets, Dynamical systems with singularities (billiards, etc.), FOS: Mathematics, Mathematics - Dynamical Systems, Other Dirichlet series and zeta functions
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