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Zeta Functions and Continuous Time Dynamics

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Modeling, Dynamics, Optimization and Bioeconomics I

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Abstract

The purpose of this survey is to introduce the reader to the relation between continuous time hyperbolic systems and zeta functions, focusing on Anosov flows and billiards as seen through the lenses of transfer operators.

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Notes

  1. 1.

    Arguably bad

  2. 2.

    The attention here is restricted to smooth manifolds, but in some occasions one could relax such hypothesis or deal with a compact metric space X.

  3. 3.

    Note that if a = 0, by transitivity, any solution f must be constant.

  4. 4.

    For example the perturbation introduced by a smooth reparametrization obtained by choosing two points which have irrational ratio.

  5. 5.

    Modulo few nontrivial caveats tied to the orientation of E s along orbits, the choice of a suitable g, the choice of how approximate the δ.

  6. 6.

    Recall that by real analytic function in a neighborhood of a point we mean that in such neighborhood the function is infinitely many time differentiable and the Taylor expansion of f(x) converges to f(x). Thus, a real-analytic manifold is a manifold such that the charts are real-analytic and a real-analytic foliation is a foliation such that the map from each leaf to \(\mathbb{R}^{n}\) is real-analytic.

  7. 7.

    C (k, α) being the class of function k-times differentiable such that the k-th derivative has Hölder exponent α.

  8. 8.

    The metric entropy is defined as h(ϕ, μ) = ∫ E u(x)d μ(x) where μ is the SRB measure related of ϕ t . See [98] for an introduction to SRB measures.

  9. 9.

    The surface of constant negative curvature constructed from a modular group.

  10. 10.

    It is an operator. Such nomenclature comes from the one-dimensional framework, where it is actually the refraction/transmission matrix associated to the potential.

  11. 11.

    By dispersing we mean that the boundary is strictly concave inward at every smooth point of the boundary. By strongly chaotic we mean hyperbolic, ergodic, mixing, and Bernoulli.

  12. 12.

    In the sense of defining det(I + A) by setting \(\det (I + A)\doteq\sum _{0}^{\infty }\text{Tr}\varLambda ^{k}(A)\).

  13. 13.

    An irrational number \(x \in \mathbb{R}\) is Diophantine if there exist ν > 0 and M > 0 such that for all \((p,q) \in \mathbb{Z} \times \mathbb{N}{\ast}\) we have \(\left \vert x -\frac{p} {q}\right \vert > \frac{M} {q^{2+\nu }}\).

  14. 14.

    One should think of this as the problem of describing the set of prime numbers such that p, p + 2 are both prime.

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Acknowledgements

I would like to thank V. Baladi, C. Liverani and M. Tsujii for helpful discussions and comments along the years. I also thank the anonymous referee for pointing out a shameful quotation error. Partially supported by ERC Advanced Grant MALADY (246953) and CNPq Brazil.

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Giulietti, P. (2014). Zeta Functions and Continuous Time Dynamics. In: Pinto, A., Zilberman, D. (eds) Modeling, Dynamics, Optimization and Bioeconomics I. Springer Proceedings in Mathematics & Statistics, vol 73. Springer, Cham. https://doi.org/10.1007/978-3-319-04849-9_18

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