Posted by: matheuscmss | January 17, 2013

Mark Pollicott’s talk on “Zeta functions for Anosov flows”

Last Monday (January 14, 2013), Mark Pollicott gave a nice talk entitled “Zeta functions for Anosov flows” at IHES. Again, the talk was attend by a highly heterogenous audience (including Emmanuel Breuillard, Maxim Kontsevich and David Ruelle) and Mark did a great job in motivating and explaining some of the main results of his long (98 pages) paper with P. Giulietti and C. Liverani (at the cost of sacrificing the presentation of proofs).

Below I will transcript my notes of Mark’s lecture. As usual, the eventual mistakes in what follows are my entire responsibility.

1. Motivations

1.1. Riemann {\zeta}-function

The Riemann {\zeta}-function “is”

\displaystyle \zeta_{Riemann}(s)=\prod\limits_{p \textrm{ prime }} (1-p^{-s})^{-1}=\sum\limits_{n=1}^{\infty}\frac{1}{n^s}

where {s\in\mathbb{C}} is a complex parameter.

Among its basic properties, we have:

  1. it converges for {\textrm{Re}(s)>1},
  2. it has a meromorphic extension to {\mathbb{C}},
  3. {s=1} is a simple pole (and it is the only one on the line {\textrm{Re}(s)=1}).

Of course, {\zeta_{Riemann}(s)} has a lot more of amazing properties (see, e.g., this list of posts by Terence Tao on this subject) and it is not our aim here to make a serious discussion of {\zeta_{Riemann}(s)}. Instead, for the sake of today’s post, let us simply recall that the analytic features of {\zeta_{Riemann}(s)} have interesting consequences into Number Theory: for example, the properties 1. and 3. above are known to imply the prime number theorem saying that

\displaystyle \frac{\#\{p\in\mathbb{N}: p \textrm{ prime}, \, p\leq N\}}{N/\log N}\rightarrow 1

as {N\rightarrow\infty}.

In other words, the study of analytic properties of Riemann’s {\zeta}-function has interesting consequences for certain counting problems in Number Theory.

Closing this subsection, let us recall that if the Riemann hypothesis is true, then the error term in the prime number theorem can be controlled in a very precise way. More concretely, if the Riemann hypothesis (claiming that {\zeta_{Riemann}(s)} has no zeroes for {1/2<\textrm{Re}(s)<1}) is true, then, for every {\varepsilon>0},

\displaystyle \#\{p\in\mathbb{N}: p \textrm{ prime}, \, p\leq N\}=Li(N)+O(N^{1/2+\varepsilon})

where {Li(N)=\int_2^N\frac{du}{\log u}}.

1.2. Selberg’s zeta function

Let V be a compact surface of genus {g\geq 2} equipped with a hyperbolic metric (i.e., a metric of negative constant curvature). Given an oriented primitive closed geodesic (also known as prime geodesic) {\tau} of {V}, let us denote its length by {\lambda(\tau)}.

The Selberg {\zeta}-function is

\displaystyle \zeta_{Selberg}(s)=\prod\limits_{\tau \textrm{ prime geodesic}}(1-e^{-s\lambda(\tau)})^{-1}

The comparison between {\zeta_{Riemann}(s)} and {\zeta_{Selberg}(s)} is usually done via the following formal dictionary: “primes” correspond to “prime geodesics”, and {p} correspond to {e^{\lambda(\tau)}}.

Among the basic properties of {\zeta_{Selberg}(s)}, we have:

  1. it converges to a non-zero analytic function for {\textrm{Re}(s)>1},
  2. it has a meromorphic extension to {\mathbb{C}},
  3. {s=1} is a simple pole (but otherwise it is non-zero and analytic on {\textrm{Re}(s)=1}),
  4. {\zeta(s)} is non-zero and analytic on {1-\varepsilon<\textrm{Re}(s)<1} for some {\varepsilon>0}.

Remark 1 This last item is a sort of “analog” of the Riemann hypothesis for {\zeta_{Selberg}(s)}.

Similarly to {\zeta_{Riemann}(s)}, the analytic features of {\zeta_{Selberg}(s)} have important consequences for the geometry of hyperbolic surfaces such as the prime geodesic theorem saying that, from properties 1. and 3. above, one has

\displaystyle \frac{\#\{\tau:\lambda(\tau)\leq T\}}{e^T/T}\rightarrow 1

as {T\rightarrow \infty}, and, from properties 1., 3. and 4. above, one has

\displaystyle \#\{\tau:\lambda(\tau)\leq T\}=Li(e^T)+O(e^{(1-\varepsilon)T})=\int_2^{e^T}\frac{du}{\log u}+O(e^{(1-\varepsilon)T})

Remark 2

  • In the literature, one usually defines Selberg’s zeta function as {Z(s)=\prod\limits_{n=0}^{\infty}\prod\limits_{\tau}(1-e^{-(s+n)\lambda(\tau)})}. This zeta function is related to the previous by the formula {\zeta_{Selberg}(s)=Z(s+1)/Z(s)} (and thus one can transfer properties from {\zeta_{Selberg}(s)} to {Z(s)} and vice-versa).
  • Closed geodesic of {V} are closed orbits for its geodesic flow {\phi_t}, and thus, the prime geodesic theorem has a dynamical meaning, namely, it solves the problem of counting closed orbits of geodesic flows on hyperbolic surfaces.

2. Results on dynamical {\zeta}-functions for Anosov flows

Let {\phi_t:M\rightarrow M} be an Anosov flow on a compact manifold {M}. As Mark Pollicott told in his lecture, if you already know the definition of Anosov flow, then it is a waste of time to repeat it, and, on the other hand, if you don’t know the definition, then showing the precise definition in a talk (or short blog post) probably will not be very inspiring.

So, instead of giving away the definition of Anosov flow, let us follow Mark Pollicott and simply mention that geodesic flows on compact manifolds {V} with negative sectional curvatures are one of the most important examples of Anosov flows.

Anyhow, let {\tau} be a closed orbit of least period {\lambda(\tau)} of the Anosov flow {\phi_t}. In this language, the Ruelle {\zeta}-function is

\displaystyle \zeta_{Ruelle}(s):=\prod\limits_{\tau}(1-e^{-s\lambda(\tau)})^{-1}

Let {h} be the topological entropy of the Anosov flow {\phi_t}. For the sake of this post, the reader can take the following formula

\displaystyle h=\lim\limits_{T\rightarrow\infty}\frac{1}{T}\log\#\{\tau: \lambda(\tau)\leq T\}

as the “definition” of the topological entropy of an Anosov flow.

Among the basic properties of Ruelle’s {\zeta}-function, we have:

  1. it converges to a non-zero analytic function for {\textrm{Re}(s)>h},
  2. it has a meromorphic extension: more precisely, one of the main results of P. Giulietti, C. Liverani and M. Pollicott is the fact that, for {C^{\infty}} Anosov flows, {\zeta_{Ruelle}(s)} has a meromorphic extension to {\mathbb{C}},
  3. {s=h} is a simple pole (and, otherwise, {\zeta_{Ruelle}(s)} is non-zero and analytic on the line {\textrm{Re}(s)=h}),
  4. it has an analytic extension slightly to the left of the line {\textrm{Re}(s)=h}: more precisely, one of the main results of P. Giulietti, C. Liverani and M. Pollicott is the fact that, for the geodesic flows on compact manifolds with {1/9}-pinched negative sectional curvatures (i.e., the ratio between the maximal and minimal sectional curvatures belongs to the interval {[1/9,1]}), {\zeta_{Ruelle}(s)} is non-zero and analytic in a small strip {h-\varepsilon<\textrm{Re}(s)<h} for some {\varepsilon>0}.

Of course, there are several comments about items 2. and 4.

Firstly, during the talk, M. Kontsevich asked what happen to item 2. in the case of Anosov flows with finite regularity, say {C^{k}}. Here, M. Pollicott said that, actually, in his paper with P. Giulietti and C. Liverani, it is shown that, in this case, {\zeta_{Ruelle}(s)} has a meromorphic extension to {Re(s)>-c(k)} where {c(k)} is a (“explicit”) constant growing linearly with {k}. In particular, for {C^{\infty}} Anosov flows, we can extend {\zeta_{Ruelle}(s)} to {\mathbb{C}}, but it is an open question whether one can do so for {C^{k}} Anosov flows in general. Here, he pointed out that the problem might be subtle because it is known that there are {C^k} Axiom A flows whose Ruelle’s zeta functions can’t be extended to {\mathbb{C}}.

Secondly, still about item 2., M. Pollicott mentioned the following previous results: in 1976, D. Ruelle showed item 2. assuming that the horocycle (i.e., stable and unstable) foliations of the Anosov flow are real-analytic ({C^{\omega}}). After that, D. Fried (following the work of H. Rugh), extended Ruelle’s result under the slightly weaker assumption that only the flow is real-analytic ({C^{\omega}}).

Thirdly, following the general philosophy about zeta functions illustrated in Section 1, one has that item 4. allows to solve a counting problem:

Corollary 1 (P. Giulietti, C. Liverani and M. Pollicott) For {1/9}-pinched negatively curved compact manifolds,

\displaystyle \#\{\textrm{closed geodesics }\tau:\,\lambda(\tau)\leq T\}=Li(e^{hT})+O(e^{(h-\varepsilon)T})

Finally, still concerning item 4. (and its corollary above), M. Pollicott mentioned the following previous results:

  • for surfaces without pinching conditions from the work of D. Dolgopyat in 1998 (previously discussed in this blog post here): indeed, even though this is not stated explicitly in Dolgopyat’s paper, it is not hard to use his methods to deduce these results.
  • the fact that the principal term in the counting problem {\#\{\textrm{closed geodesics }\tau:\,\lambda(\tau)\leq T\}} is

    \displaystyle e^{hT}/(hT)\sim Li(e^{hT})

    was known from the results in G. Margulis’ PhD thesis in 1970.

  • actually, P. Guilietti, C. Liverani and M. Pollicott show their results for “contact Anosov flows satisfying certain {1/3}-bunching conditions”, a class of flows including “geodesic flows on {1/9}-pinched negatively curved manifolds”.

As the time was running short, M. Pollicott said just a few words about the proofs of these results.

Very roughly speaking, a fundamental idea in the works of D. Ruelle and others on this subject is the fact that {\zeta_{Ruelle}(s)} is a sort of determinant of certain (transfer-like) operators in adequate Banach spaces. Here, the “quality” of the Banach space one chooses to work with affects directly the “quality” of the properties of {\zeta_{Ruelle}(s)} one can deduce. In particular, D. Ruelle and D. Fried imposed a severe ({C^{\omega}}) regularity assumption on the Anosov flow precisely to be able to make a “comfortable” choice of Banach space.

At this point, the main novelty in the work of P. Giulietti, C. Liverani and M. Pollicott is the observation that, in recent years, after the works of V. Baladi, S. Goüezel, C. Liverani and M. Tsujii, our understanding of anisotropic Banach spaces adapted to Anosov (hyperbolic) systems is considerably advanced. In particular, using these works as a guideline, P. Giulietti, C. Liverani and M. Pollicott choose Banach spaces adapted to “less smooth” Anosov flows in the sense that {\zeta_{Ruelle}(s)} is “written” as a determinant of a certain operator in such a Banach space.

Closing his talk, M. Pollicott briefly mentioned it is almost impossible to discuss {\zeta_{Ruelle}(s)} without talking about decay of correlations for the measure of maximal entropy (the reason being that closed orbits of Anosov flows equidistribute to the maximal entropy measure), and, in this direction, he mentioned the following results.

Let {\mu} be the maximal entropy measure of the Anosov flow {\phi_t:M\rightarrow M}. Given {F, G\in C^{\infty}(M)}, let us denote the correlation function by

\displaystyle \rho(t):=\int (F\circ \phi_t) G \,d\mu - \int F\,d\mu\,\int G\,d\mu

In general, the asymptotic behavior of {\rho(t)} (i.e., the rate of mixing/decay of correlations) is described by the analytic properties of the Fourier/Laplace transform

\displaystyle \widehat{\rho}(s)=\int_0^{\infty}e^{-st}\rho(t)\,dt

for {s\in\mathbb{C}}.

Here, it is shown by P. Giulietti, C. Liverani and M. Pollicott that:

  • {\widehat{\rho}(s)} converges to an analytic function for {\textrm{Re}(s)>0},
  • {\widehat{\rho}(s)} has a meromorphic extension to {\mathbb{C}} (actually, this is essentially known since this work of O. Butterley and C. Liverani here),
  • morally, “if {s} is a pole of {\widehat{\rho}}, then {s+h} is a pole for {\zeta_{Ruelle}(s)}” (this is not always true, but almost…),
  • there are {C, \lambda>0} such that {|\widehat{\rho}(t)|\leq Ce^{-\lambda t}}, {t>0}, providing geodesic flows on {1/9}-pinched negatively curved manifolds.

After this last comment, the talk was over and several questions were posed to M. Pollicott. Closing this post, I will reproduce here just two questions (together with their respective answers):

  • Maxim Kontsevich asked about results for other {\zeta}-functions where we “twist” {\zeta_{Ruelle}(s)} with some weights (i.e., replace lengths {\lambda(\tau)} by “top Lyapunov exponent”) and/or {L}-versions (i.e., replace lenght by characters associated to the homology). Here, M. Pollicott told that he believes that it might be possible to extend the current results to these settings, but, since his article with P. Guilietti and C. Liverani is long, such extensions might demand a lot of extra work and this is why they have not attempt to discuss this in their article. Also, M. Pollicott told that lengths of geodesics are dynamically natural because closed geodesics equidistribute to the maximal entropy measure, but, if we start twisting, we get a measure probably coming from thermodynamical formalism whose “geometry” we might not control as well as the maximal entropy and this could create more technical difficulties.
  • Emmanuel Breuillard asked whether {\zeta_{Ruelle}(s)} admits functional equations in general (similarly to the case of Riemann’s {\zeta}-function, Selberg’s {\zeta}-function on the modular curve and, more generally, in contexts related to Number Theory). Here, M. Pollicott said that the “natural” answer would be “no”: indeed, in some sense, the “explanation” for the presence of functional equations in number-theoretical contexts is “related” to the fact that the underlying (“arithmetic”) dynamical systems belong to a some finite-dimensional Teichmüller/moduli spaces (i.e., a relatively “small” parameter space), but, in general, Anosov flows (such as geodesic flows in manifolds with variable negative curvature) tend to live in infinite dimensional Teichmüller/moduli spaces and thus the existence of functional equations would be somewhat surprising.


  1. the first link is wrong.

    • Corrected, thanks!

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