Bowen-Margulis measure

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The Bowen-Margulis measure is an invariant measure for a hyperbolic dynamical system. It is named after Rufus Bowen and Gregory Margulis, who gave two completely different constructions of it. Both constructions give a measure of maximal entropy, and this is unique for the systems under consideration, hence these constructions yield the same measure.


Maximizing entropy

For a dynamical system (a map or a flow) that preserves a probability measure one can define the measure-theoretic or Kolmogorov-Sinai entropy, a measure of the degree of unpredictability in the system measured on an exponential scale (Katok, Hasselblatt 1995, p. 169). For a topological dynamical system on a compact space topological entropy provides a topological counterpart. Since topological dynamical systems on compact spaces have an invariant Borel probability measure, both notions of entropy are defined in this case, and there may be a choice of invariant measures and, accordingly, various possible values of measure-theoretic entropy. The topological entropy is an upper bound for these (Goodwyn 1969), and the Variational Principle (Katok, Hasselblatt 1995, p. 181) says that for homeomorphisms of compact metric spaces it is the least upper bound (Dinaburg for finite topological dimension and Goodman in general). While the supremum may not be attained in all case, some proofs of the Variational Principle indeed show that if the homeomorphism is expansive then the supremum is indeed attained, i.e., there is a measure of maximal entropy (see, for instance, Katok, Hasselblatt 1995, p. 182). Finally, Bowen showed that if one furthermore assumes his specification property (which holds for a compact topologically transitive hyperbolic set) then this measure of maximal entropy is indeed unique (Katok, Hasselblatt 1995, p. 620).

Margulis measure

In his 1970 dissertation Margulis (Margulis 1970) introduced the Margulis measure for an Anosov flow by a local weighted product of measures on stable and unstable leaves, and these measures were chosen such as to scale under the dynamics by a factor of \(e^{ht}\ ,\) where \(h\) is the topological entropy. They are also invariant under strong (un)stable holonomies.

Specifically, if \(\mu^{0u}\) denotes this measure on weak unstable leaves then \[\mu^{0u}(\varphi^t(U))=e^{ht}\mu^{0u}(U)\] for every measurable set \(U\) in a weak-unstable leaf, and analogously, \[\mu^{0s}(\varphi^t(U))=e^{-ht}\mu^{0u}(U)\ .\]

The Margulis measure is then locally defined essentially as the product measure \(\mu^{0u}\times\mu^s\ .\)

Margulis proved that this measure has strong mixing properties.

The measures \(\mu^{0u}\) and \(\mu^{0s}\) are characterized also by holonomy-invariance, that is, if \(U\) is a compact subset of a weak-unstable leaf \(W^{0u}(x)\ ,\) and \(U'\) is a subset of a nearby leaf \(W^{0u}(y)\) that is obtained from \(U\) by moving every point of \(U\) to \(U'\) along a strong-stable leaf (i.e., \(U'=\bigcup_{z\in U}W_{\hbox{loc}}^s(z)\cap W^{0u}(x')\)), then \(\mu^{0u}(U')=\mu^{0u}(U)\ .\) This property that has been found to characterize them (Bowen, Marcus 1977).

Bowen measure

In hyperbolic dynamics the topological entropy coincides with the exponential growth rate of the number of periodic orbits as a function of their lengths. Bowen sharpened this insight with his Specification Theorem, which says, roughly, that any conceivable sequence of orbit segments in a topologically mixing hyperbolic set is realized to arbitrary accuracy by a periodic orbit. This causes topological entropy to be represented by periodic points, and accordingly, his construction is based on measures carried by periodic orbits (Bowen 1971, Bowen 1972).

Suppose \(f\) is a continuous map for which \(\operatorname{card}\,\operatorname{Fix}(f^n)<\infty\) for all \(n\in\mathbb{N}\) (this is true for expansive homeomorphisms), and denote by \(\delta_x\) the probability measure concentrated on \(x\ .\) The Bowen measure is defined by \[\mu_{\operatorname{BMM}}:=\lim_{n\to\infty}\sum_{x\in\operatorname{Fix}(f^n)}\delta_x\big/\operatorname{card}\,\operatorname{Fix}(f^n)\ ,\]

provided, the limit exists. This is indeed the case in the presence of the specification property. In fact, Bowen first considered any weak*-accumulation point \(\mu\) of \(\sum_{x\in\operatorname{Fix}(f^n)}\delta_x\big/\operatorname{card}\,\operatorname{Fix}(f^n)\ .\) Using the specification property one shows that if \(\nu\) is a measure of maximal entropy (which exists by expansivity) then \(\nu=\mu\ ,\) which establishes uniqueness and in particular existence of the limit. It is in this latter situation that it makes sense to name the measure after Bowen.

The Patterson-Sullivan Construction

The Margulis measure is constructed by producing the measures \(\mu^{0u}\) and \(\mu^{0s}\) as well as closely related measures \(\mu^u\) and \(\mu^s\) on leaves of the invariant foliations. The connection is that \(d\mu^{0u}(\varphi^t(y))=e^{ht}dtd\mu^{0u}\ .\)

In the case of geodesic flows there is an important way of viewing these measures. On the universal cover \(\tilde M\) of the Riemannian manifold \(M\ ,\) each (strong) stable or unstable leaf is canonically identified with the punctured sphere at infinity by following geodesics to infinity, for instance, by the bijection \[P_x\colon W^s(x)\to\tilde M(\infty)\smallsetminus\{c_x(\infty)\}\] defined by \(P_x(z)=c_z(-\infty)\ .\)

Accordingly, the measures \(\mu^u\) on the translates of a strong stable leaf by the geodesic flow produce a collection of pairwise proportional measures at infinity.

The Patterson-Sullivan construction is based on the Poincaré series \[P(s,p,q):=\sum_{\gamma\in\pi_1(M)}e^{-sd(p,\gamma q)}\ ,\] which converges for \(s>h\) and diverges for \(s\le h\ ,\) where \(h\) is again the topological entropy of the geodesic flow. For \(s>h\) one then defines \[\mu_{p,q,s}:=\frac{\sum_{\gamma\in\pi_1(M)}e^{-sd(p,\gamma q)}\delta_{\gamma q}}{P(s,q,q)}\ ,\] where \(\delta_x\) is the Dirac \(\delta\)-measure concentrated on \(x\ .\) Similarly to what happens in the Bowen construction one obtains a measure \(\nu_p:=\lim_{s\to h}\mu_{p,q,s}\) (weak limit) which is unique (and hence independent of \(q\)). These are the Patterson-Sullivan measures, and they scale as \(e^{ht}\) under the flow, so by uniqueness, these are the conditionals of the Margulis measure.

Patterson first constructed these measures on the limit set of a Fuchsian group, in particular, in constant curvature. The idea was further developed by Sullivan (still in constant curvature), and Kaimanovich (1990) applied it to manifolds of nonconstant negative curvature and established the connection with the Bowen-Margulis measure (by verifying the scaling property described above; a more direct proof is in Kaimanovich 1991).

Hausdorff Measures

Yet another description of the Margulis measure has some similarities to the Patterson-Sullivan construction, but applies to all Anosov flows. This is a description due to Hamenstädt of the conditionals of the Margulis measure on strong-unstable (and likewise strong-stable) leaves as a Hausdorff measure associated with a dynamically defined distance.

The underlying distance on a leaf \(W^u(z)\) is defined by fixing some \(R>0\) and setting \[\eta(x,y):=e^{-\sup\{t\in\mathbb{R}\mid d^u(\varphi^t(x),\varphi^t(y))\le R}\ .\] Here, \(d^u\) is the distance inside an unstable leaf that is induced by the Riemannian metric.

Note that \(\eta\circ\varphi^t=e^t\eta\ ,\) \(\eta\ge0\ ,\) \(\eta(x,y)=\eta(y,x)\) and \(\eta(x,y)=0\) if and only if \(x=y\ .\) Furthermore, one can show that \[\frac1C(\eta(x,z)^a\le(\eta(x,y))^a+(\eta(y,z))^a\] whenever \(x,y,z\) are in the same strong-unstable leaf. Here \(a:=-\log\lambda>0\ ,\) and \(C\) and \(\lambda\) are as in the definition of an Anosov flow. Thus, if we choose a Lyapunov metric (a Riemannian metric such that \(C=1\) in the definition of an Anosov flow) then \(\eta^a\) defines a distance on each strong-unstable leaf. (For a geodesic flow the standard metric on the unit tangent bundle is a Lyapunov metric.)

For a transitive Anosov flow (and using a Lyapunov metric), the strong-unstable conditionals of the Margulis measure are, up to normalization, given by the \(h/a\)-dimensional Hausdorff measure associated with the distance \(\eta^a\) where \(h\) is again the topological entropy. These measures are, specifically, \[\mu_H(A)=\sup_{\epsilon>0}\inf\{\sum_i\hbox{diam}(S_i)^h\mid A\subset\bigcup_{i=0}^\infty S_i, \hbox{diam}(S_i)\le\epsilon\}\ .\]

One could instead also use the \(h/a\)-dimensional spherical measure associated with the distance \(\eta^a\ ,\) which is \[\mu_S(A)=\sup_{\epsilon>0}\inf\{\sum_ir_i^h\mid A\subset\bigcup_{i=0}^\infty B_\eta(x_i,\epsilon_i), \epsilon_i\le\epsilon\}\ .\]

Periodic Orbits

The introduction of the Margulis measure is only part of the dissertation of Margulis. Another part is the application of this construction to give a precise asymptotic formula for the growth rate of the number of periodic orbit of an Anosov flow: the number \(N(T)\) of periodic orbits of length at most \(T\) is asymptotic to \(e^{hT}/hT\) as \(T\to\infty\) in the sense that \[N(T)hTe^{-hT}\to1\] as \(T\to\infty\ .\) The best previous asymptotic estimate was that the left-hand side is between two positive constants.

(Margulis also produced a related asymptotic of the number of lattice points or geodesic arcs: Given any \(x\ ,\) \(y\) in the universal cover the number of \(\gamma\) in the fundamental group for which \(d(x,\gamma y)\le T\) is asymptotically \(Ce^{hT}\ .\))

Intuitively, this is obtained from the fact that the scaling property of the Margulis measure gives precise information about the measure of flow boxes, and since it coincides with the Bowen measure, this is related to precise information about numbers of periodic points.

This growth asymptotic extends to hyperbolic flows that are not necessarily Anosov, and there are various more recent refinements: If a negatively curved Riemannian manifold has dimension 2 or is 1/4-pinched then the geodesic flow satisfies \[N(T)=\hbox{li}(e^{hT})+O(e^{cT})\] for some \(c<h\ ,\) where \(\hbox{li}(x)=\int_2^x1/\log(s)ds\ ,\) and an analogous result holds for some dispersing planar billiards. More generally, \[N(T)=\hbox{li}(e^{hT})(1+O(T^{-\delta}))\] for some \(\delta>0\) in the case of a weakly mixing flow that is either transitive Anosov or hyperbolic with three periodic orbits \(\gamma_i\) for which \[\frac{l(\gamma_1)-l(\gamma_2)}{l(\gamma_2)-l(\gamma_3)}\] is badly approximable. These results are proved using zeta-function techniques.

The construction of the Margulis measure was quite recently extended to geodesic flows of nonpositively curved Riemannian manifolds of rank 1 using the Patterson-Sullivan construction. This leads to an extension of the asymptotic to these weakly hyperbolic flows: If one defines \(N(T)\) for geodesic flows of nonpositively curved manifolds by counting modulo free homotopy, and \(N_{\hbox{reg}}(T)\) counts only regular geodesics then \[e^{hT}/(CT)\le N_{\hbox{reg}}(T)\le N(T)\le Ce^{hT}\] for some \(C>1\ ,\) and \[N(T)-N_{\hbox{reg}}(T)=O(e^{(h-\epsilon)T})\] for some \(\epsilon>0\ .\)

Generalizations of the Bowen Construction

The concept of entropy can be generalized to that of pressure, a notion that also has topological and measure-theoretic versions. The definition of topological entropy (as recast by Bowen) is based on counting orbit segments, and the notion of topological pressure modifies this notion by assigning a weight to each point of the phase space. This is done by taking a Hölder continuous function \(\psi\) on the phase space and attaching to each point \(x\) the weight \(e^{\psi(x)}\ .\) Topological pressure is then defined in the same manner as topological entropy, except that sums over separated or spanning sets of orbit segments are replaced by the corresponding weighted sums. Specifically, each point \(x\) of an \((n,\epsilon)\)-separated set of a map \(f\) will be counted with weight \(e^{\sum_{i=0}^{n-1}\psi(f^i(x))}\ .\)

The measure-theoretic counterpart is simply defined as measure-theoretic entropy plus the space average of the potential \(\psi\ .\) (The terminology has motivations in statistical physics.)

There is a corresponding variational principle: For a given potential \(\psi\) there is a unique measure of maximal pressure. This counterpart of the Bowen-Margulis measure is referred to as the equilibrium state for \(\psi\ .\)

Bowen (1974) established uniqueness of equilibrium states in expansive systems with specification for functions \(\psi\) with the following property: There exists an \(\epsilon>0\) and a \(K\) such that if \(d(f^k(x),f^k(y))\leq\epsilon\) for \(0\leq k<n\) then

\[|\sum_{i=0}^{n-1}\psi(f^i(x))-\sum_{i=0}^{n-1}\psi(f^i(y))|\leq K\ .\]


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