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I am reading a theorem about the ergodicity and mixing of geodesic and horocyclic flows on unit tangent bundle of a compact hyperbolic surface. I find that there are two ways (be listed below) to approach that theorem.

First way:

Step 1. Define Liouville measure on unit tangent bundle which is the measure product of Lebesgue measure on $\mathbb{S}_{1}$ and hyperbolic measure on surface. By a evenly cover mapping, the hyperbolic measure on surface comes from hyperbolic area in upper plane model.

Step 2. Prove that geodesic flows and horocyclic flows preserves that measure.

Step 3. Use representation theory and operator analysis to prove the ergodicity and mixing of that flows.

Second way:

Step 0. Identify unit tangent bundle with the quotient space $\Gamma\setminus SL(2,\mathbb{R})$, where $\Gamma$ is a symmetric discrete subgroup of $SL(2,\mathbb{R})$.

Step 1. Define the measure of SL(2,R) be measure product of its subgroups K, A, N.

Step 2. Work on SL(2,R) which is the covering space of the quotient space. Prove that geodesic flows and horocyclic flows (be written in SL(2,R) - coordinates) preserves the measure in step 1.

Step 3. Use representation theory and operator analysis to prove the ergodicity and mixing of that flows.

My question:

  1. Are these above steps exact ?

  2. Is there another way to approach that theorem ?

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  • $\begingroup$ Step 0-1-2 are redundant in my opinion (everyone learns about mobius transformations and action of SL2 on the upper half plane in undergrad group theory/complex analysis class). The representation theoretical (Howe-Moore thm, Mautner phenomenon and extensions by HC etc) approach is highly fruitful (as it leads to quantitative estimates), but historically it is not the first one to be used. The first argument (by Hopf) using just the Anosov property of the geodesic flow, hence generalizing to other situation, and you can probably refine this argument to get mixing $\endgroup$ – Asaf Mar 26 '17 at 18:56
  • $\begingroup$ (at least basic mixing, I'm not sure about mixing of all orders). This apply to the geodesic flow, The argument for the horocyclic flow follows by the re-normalization relations (this whats goes into Mautner's proof by the way). $\endgroup$ – Asaf Mar 26 '17 at 18:56
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    $\begingroup$ @Asaf not every undergrad group theory class is like the one you are thinking of (same goes for complex analysis) $\endgroup$ – Yemon Choi Mar 26 '17 at 21:13
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The algebraic approach is just one of many ways to deal with the geodesic and horocycle flows. Also your quick summary does not really pay tribute to the many ways to deal with it using "representation theory and operator analysis".

The geodesic flow can be defined physically, associated to curves with zero acceleration (Newton), as critical points of some action (Lagrange), or obtained from the energy functional (Hamilton, Jacobi). This corresponds to the free movement of a particle on a compact metric space of constant curvature. The metric itself can be defined following Gauss by building isothermal coordinates on the surface, which itself can be done in several ways, one of which consisting in solving some parabolic EDP, and then any self-respecting analysts will want to do it in many interesting ways.

There is a geometric definition using the Poincaré disk or the Poincaré upper half plane or some other model in hyperbolic geometry, either by taking a quotient or starting from the surface itself and using some uniformization procedure of the universal cover à la Hadamard, or maybe the classification of simply connected Riemann surfaces which admits several proofs by different methods, the original one by Riemann dealing with electromagnetism, the Laplacian and harmonic functions. Here again, the metric and the geodesic flow can be defined infinitesimally, at finite range or using a boundary at infinity.

It can be defined algebraically, as you alluded to, using $SL_2({\bf R})$, $PSU_{1,1}({\bf R})$ or the group of translation and homotheties of the real line or some other groups I can't remember, each of them being identified with the isometry group of the geometric models previously discussed. The geodesic flow is then defined as a matrix action, or maybe just as a one-dimensional Lie subgroup using its infinitesimal generator.

Of course there is also a symbolic representation of the geodesic flow. A very interesting example is given by the (non-compact) modular surface and the decomposition of real numbers in continuous fractions. There are several ways to build such models from a geometric or algebraic representation, cutting sequences, closed geodesics, etc.

And now we come to the properties of the geodesic and horocyclic flows which are numerous and in no way restricted to ergodicity and mixing. E. Hopf himself gave several proofs of the ergodicity of the geodesic flow, the first one using holomorphic functions, harmonic analysis and the Harnack principle. He is more often remembered for a dynamic proof known as the Hopf argument which uses the hyperbolic nature of the flow. Hedlund proved the mixing property circa 1939. D.V. Anosov generalized these results to the variable curvature case and invented Anosov flows at the same time. Interestingly, the proof of mixing by Anosov and Sinai uses the symplectic nature of the flow, together with a bit of cohomology, and the then all new entropy theory.

The algebraic approach starts with Gelfand, then Mautner. The unique ergodicity of the horocyclic flow is due to Furstenberg using technics from harmonic analysis/representation theory again, and a parallel can be made with rotations of the circle. This is followed by a dynamic proof of Marcus, and then the symbolic work of Bowen, Marcus. A famous generalization to unipotent actions on Lie group is due to Ratner, 1984, and builds on the dynamic approach of Anosov and Sinai using entropy theory, Lyapunov exponents and more. The work of Howe and Moore, 1977, study the correlations through representation theory, which is perhaps the work you are alluding to. Moore 1987 also gives the exponential mixing of the geodesic flow. This is now extended to symplectic Anosov flows using improved symbolic technics by Dolgopyat or anisotropic Banach spaces and spectral gaps by Liverani and a number of coauthors.

I have only scratched the surface here. I could cite at least ten books with different proofs of the mixing of the geodesic flow, dozens authors, a hundred papers and there are proofs published almost every years, since a century.

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    $\begingroup$ Regarding the history of "the algebraic approach": it seems that representation theory was first used for this problem by Gelfand-Fomin ['52] in dimension 2 and 3. Mautner ['57] generalized this to an arbitrary dimension, using a tool called nowadays "Mautener phenomenon", which in fact is a lemma appearing already in a paper by Segal-von-Neumann ['50]. More on the subject in the first section of arxiv.org/pdf/1408.4217.pdf. $\endgroup$ – Uri Bader Mar 27 '17 at 9:45

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