Bowen-Margulis Measure Research Articles

Stable random fields, Patterson–Sullivan measures and extremal cocycle growth

We study extreme values of group-indexed stable random fields for discrete groups G acting geometrically on spaces X in the following cases: (1) G acts properly discontinuously by isometries on a CAT(-1) space X, (2) G is a lattice in a higher rank Lie group, acting on a symmetric space X, and (3) G is the mapping class group of a surface acting on its Teichmüller space. The connection between extreme values and the geometric action is mediated by the action of the group G on its limit set equipped with the Patterson–Sullivan measure. Based on motivation from extreme value theory, we introduce an invariant of the action called extremal cocycle growth which measures the distortion of measures on the boundary in comparison to the movement of points in the space X and show that its non-vanishing is equivalent to finiteness of the Bowen–Margulis measure for the associated unit tangent bundle U(X/G) provided X/G has non-arithmetic length spectrum. As a consequence, we establish a dichotomy for the growth-rate of a partial maxima sequence of stationary symmetric \(\alpha \)-stable (\(0< \alpha < 2\)) random fields indexed by groups acting on such spaces. We also establish analogous results for normal subgroups of free groups.

Equilibrium states for self‐products of flows and the mixing properties of rank 1 geodesic flows

Equilibrium states for geodesic flows over closed rank 1 manifolds were studied recently by Burns, Climenhaga, Fisher and Thompson. For sufficiently regular potentials, it was shown that if the singular set does not carry full pressure then the equilibrium state is unique. The main result of this paper is that these equilibrium states have the Kolmogorov property. In particular, these measures are mixing of all orders and have positive entropy. For the Bowen-Margulis measure, we go further and obtain the Bernoulli property from the K-property using classic arguments from Ornstein theory. Our argument for the K-property is based on an idea due to Ledrappier. We prove uniqueness of equilibrium states on the product of the system with itself. To carry this out, we develop techniques for uniqueness of equilibrium states which apply in the presence of the 2-dimensional center direction which appears for a product of flows. This is a key technical challenge of this paper.

Equidistribution of horospheres in non-positive curvature

AbstractWe study the ergodic properties of horospheres on rank 1 manifolds with non-positive curvature. We prove that the horospheres are equidistributed under the action of the geodesic flow towards the Bowen–Margulis measure, on a large class of manifolds. In the case of surfaces, we define a parametrization of the horocyclic flow on the set of horocycles containing a rank 1 vector that is recurrent under the action of the geodesic flow. We prove that the horocyclic flow in restriction to this set is uniquely ergodic. The results are valid for large classes of manifolds, including the compact ones.

A Note on the Relation Between the Metric Entropy and the Generalized Fractal Dimensions of Invariant Measures

We investigate in this work some situations where it is possible to estimate or determine the upper and the lower q-generalized fractal dimensions $$D^{\pm }_{\mu }(q)$$ , $$q\in {\mathbb {R}}$$ , of invariant measures associated with continuous transformations over compact metric spaces. In particular, we present an alternative proof of Young’s Theorem by Young (Ergod. Theory Dyn. Syst. 2(1):109–124, 1982) for the generalized fractal dimensions of the Bowen-Margulis measure associated with a $$C^{1+\alpha }$$ -Axiom A system over a two-dimensional compact Riemannian manifold M. We also present estimates for the generalized fractal dimensions of an ergodic measure for which Brin-Katok’s Theorem is pointwise satisfied, in terms of its metric entropy. Furthermore, for expansive homeomorphisms (like $$C^1$$ -Axiom A systems), we show that the set of invariant measures such that $$D_\mu ^+(q)=0$$ ( $$q\ge 1$$ ), under a hyperbolic metric, is generic (taking into account the weak topology). We also show that for each $$s\in [0,1)$$ , $$D^{+}_{\mu }(s)$$ is bounded above, up to a constant, by the topological entropy, also under a hyperbolic metric. Finally, we show that, for some dynamical systems, the metric entropy of an invariant measure is typically zero, settling a conjecture posed by Sigmund (Trans. Am. Math. Soc. 190:285–299, 1974) for Lipschitz transformations which satisfy the specification property.

Limiting distribution of geodesics in a geometrically finite quotients of regular trees

Let \mathcal T be a (q+1) -regular tree and let \Gamma be a geometrically finite discrete subgroup of the group \operatorname{Aut}(\mathcal T) of automorphisms of \mathcal T . In this article, we prove an extreme value theorem on the distribution of geodesics in a non-compact quotient graph \Gamma\backslash\mathcal{T} . Main examples of such graphs are quotients of a Bruhat–Tits tree by non-cocompact discrete subgroups \Gamma of \operatorname{PGL}(2,\mathbf{K}) of a local field \mathbf{K} of positive characteristic. We investigate, for a given time T , the measure of the set of \Gamma -equivalent classes of geodesics with distance at most N(T) from a sufficiently large fixed compact subset D of \Gamma\backslash\mathcal{T} up to time T . We show that there exists a function N(T) such that for Bowen–Margulis measure \mu on the space \Gamma\backslash\mathcal{GT} of geodesics and the critical exponent \delta of \Gamma , \lim_{T\to\infty}\mu(\{[l]\in\Gamma\backslash\mathcal{GT}\colon \underset{0\le t \le T}{\textrm{max}}d(D,l(t))\le N(T)+y\})=e^{-q^y/e^{2\delta y}}. In fact, we obtain a precise formula for N(T) : there exists a constant C depending on \Gamma and D such that N(T)=\log_{e^{2\delta/q}}\Big(\frac{T(e^{2\delta-q)}}{2e^{2\delta}-C(e^{2\delta}-q)}\Big).

The unique measure of maximal entropy for a compact rank one locally CAT(0) space

<p style='text-indent:20px;'>Let <inline-formula><tex-math id="M1">\begin{document}$ X $\end{document}</tex-math></inline-formula> be a compact, geodesically complete, locally CAT(0) space such that the universal cover admits a rank one axis. We prove the Bowen-Margulis measure on the space of geodesics is the unique measure of maximal entropy for the geodesic flow, which has topological entropy equal to the critical exponent of the Poincaré series.

Strong symbolic dynamics for geodesic flows on CAT(-1) spaces and other metric Anosov flows

We prove that the geodesic flow on a locally CAT(-1) metric space which is compact, or more generally convex cocompact with non-elementary fundamental group, can be coded by a suspension flow over an irreducible shift of finite type with Hölder roof function. This is achieved by showing that the geodesic flow is a metric Anosov flow, and obtaining Hölder regularity of return times for a special class of geometrically constructed local cross-sections to the flow. We obtain a number of strong results on the dynamics of the flow with respect to equilibrium measures for Hölder potentials. In particular, we prove that the Bowen-Margulis measure is Bernoulli except for the exceptional case that all closed orbit periods are integer multiples of a common constant. We show that our techniques also extend to the geodesic flow associated to a projective Anosov representation [BCLS15], which verifies that the full power of symbolic dynamics is available in that setting.

Equidistribution and counting of orbit points for discrete rank one isometry groups of Hadamard spaces

Let $X$ be a proper, geodesically complete Hadamard space, and $\ \Gamma<\mbox{Is}(X)$ a discrete subgroup of isometries of $X$ with the fixed point of a rank one isometry of $X$ in its infinite limit set. In this paper we prove that if $\Gamma$ has non-arithmetic length spectrum, then the Ricks' Bowen-Margulis measure -- which generalizes the well-known Bowen-Margulis measure in the CAT$(-1)$ setting -- is mixing. If in addition the Ricks' Bowen-Margulis measure is finite, then we also have equidistribution of $\Gamma$-orbit points in $X$, which in particular yields an asymptotic estimate for the orbit counting function of $\Gamma$. This generalizes well-known facts for non-elementary discrete isometry groups of Hadamard manifolds with pinched negative curvature and proper CAT$(-1)$-spaces.

Ergodicity of Bowen–Margulis measure for the Benoist 3-manifolds

<p style="text-indent:20px;">We study the geodesic flow of a class of 3-manifolds introduced by Benoist which have some hyperbolicity but are non-Riemannian, not CAT(0), and with non-<inline-formula><tex-math id="M1">\begin{document}$ C^1 $\end{document}</tex-math></inline-formula> geodesic flow. The geometries are nonstrictly convex Hilbert geometries in dimension three which admit compact quotient manifolds by discrete groups of projective transformations. We prove the Patterson–Sullivan density is canonical, with applications to counting, and construct explicitly the Bowen–Margulis measure of maximal entropy. The main result of this work is ergodicity of the Bowen–Margulis measure.

On the geodesic flow on CAT(0) spaces

Under certain assumptions on CAT(0) spaces, we show that the geodesic flow is topologically mixing. In particular, the Bowen–Margulis’ measure finiteness assumption used by Ricks [Flat strips, Bowen–Margulis measures, and mixing of the geodesic flow for rank one CAT(0) spaces. Ergod. Th. &amp; Dynam. Sys. 37 (2017), 939–970] is removed. We also construct examples of CAT(0) spaces that do not admit finite Bowen–Margulis measure.

Ergodic properties of some negatively curved manifolds with infinite measure

Let $M=X/\Gamma$ be a geometrically finite negatively curved manifold with fundamental group $\Gamma$ acting on $X$ by isometries. The purpose of this paper is to study the mixing property of the geodesic flow on $T^1M$, the asymptotic equivalent as $R\longrightarrow+\infty$ of the number of closed geodesics on $M$ of length less than $R$ and of the orbital counting function $\sharp\{\gamma\in\Gamma | d(o,\gamma.o)\le R\}$. These properties are well known when the Bowen-Margulis measure on $T^1M$ is finite. We consider here divergent Schottky groups whose Bowen-Margulis measure is infinite and ergodic, and we precise these ergodic properties using a suitable symbolic coding.

Finiteness of Gibbs measures on noncompact manifolds with pinched negative curvature

We characterize the finiteness of Gibbs measures for geodesic flows on negatively curved manifolds by several criteria, analogous to those proposed by Sarig for symbolic dynamical systems over an infinite alphabet. As an application, we recover Dal'bo-Otal-Peigne criterion of finiteness for the Bowen-Margulis measure on geometrically finite hyperbolic manifolds, as well as Peigne' examples of gemetrically infinite manifolds having a finite Bowen-Margulis measure. These criteria should be useful in the future to find more examples with finite Gibbs measures.

Asymptotic geometry of negatively curved manifolds of finite volume

We study the asymptotic behaviour of simply connected, Riemannian manifolds X of strictly negative curvature admitting a non-uniform lattice Γ. If the quotient manifold ¯ X = Γ\X is asymptotically 1/4-pinched, we prove that Γ is divergent and U ¯ X has finite Bowen-Margulis measure (which is then ergodic and totally conservative with respect to the geodesic flow); moreover, we show that, in this case, the volume growth of balls B(x, R) in X is asymptotically equivalent to a purely exponential function c(x)e δR , where δ is the topological entropy of the geodesic flow of ¯ X. This generalizes Margulis' celebrated theorem to negatively curved spaces of finite volume. In contrast, we exhibit examples of lattices Γ in negatively curved spaces X (not asymptotically 1/4-pinched) where, depending on the critical exponent of the parabolic subgroups and on the finiteness of the Bowen-Margulis measure, the growth function is exponential, lower-exponential or even upper-exponential.

Flat strips, Bowen–Margulis measures, and mixing of the geodesic flow for rank one CAT(0) spaces

Let$X$be a proper, geodesically complete CAT($0$) space under a proper, non-elementary, isometric action by a group$\unicode[STIX]{x1D6E4}$with a rank one element. We construct a generalized Bowen–Margulis measure on the space of unit-speed parametrized geodesics of$X$modulo the$\unicode[STIX]{x1D6E4}$-action. Although the construction of Bowen–Margulis measures for rank one non-positively curved manifolds and for CAT($-1$) spaces is well known, the construction for CAT($0$) spaces hinges on establishing a new structural result of independent interest: almost no geodesic, under the Bowen–Margulis measure, bounds a flat strip of any positive width. We also show that almost every point in$\unicode[STIX]{x2202}_{\infty }X$, under the Patterson–Sullivan measure, is isolated in the Tits metric. (For these results we assume the Bowen–Margulis measure is finite, as it is in the cocompact case.) Finally, we precisely characterize mixing when$X$has full limit set: a finite Bowen–Margulis measure is not mixing under the geodesic flow precisely when$X$is a tree with all edge lengths in$c\mathbb{Z}$for some$c&gt;0$. This characterization is new, even in the setting of CAT($-1$) spaces. More general (technical) versions of these results are also stated in the paper.

Skinning measures in negative curvature and equidistribution of equidistant submanifolds

AbstractLet$C$be a locally convex closed subset of a negatively curved Riemannian manifold$M$. We define the skinning measure${\sigma }_{C} $on the outer unit normal bundle to$C$in$M$by pulling back the Patterson–Sullivan measures at infinity, and give a finiteness result for${\sigma }_{C} $, generalizing the work of Oh and Shah, with different methods. We prove that the skinning measures, when finite, of the equidistant hypersurfaces to$C$equidistribute to the Bowen–Margulis measure${m}_{\mathrm{BM} } $on${T}^{1} M$, assuming only that${m}_{\mathrm{BM} } $is finite and mixing for the geodesic flow. Under additional assumptions on the rate of mixing, we give a control on the rate of equidistribution.

Proper isometric actions of hyperbolic groups on -spaces

AbstractWe show that every non-elementary hyperbolic group $\G $ admits a proper affine isometric action on $L^p(\bd \G \times \bd \G )$, where $\bd \G $ denotes the boundary of $\G $ and $p$ is large enough. Our construction involves a $\G $-invariant measure on $\bd \G \times \bd \G $ analogous to the Bowen–Margulis measure from the ${\rm CAT}(-1)$ setting, as well as a geometric, Busemann-type cocycle. We also deduce that $\G $ admits a proper affine isometric action on the first $\ell ^p$-cohomology group $H^1_{(p)}(\G )$ for large enough $p$.

Bowen-Margulis measure

Bowen-Margulis measure

Dynamique sur le rayon modulaire et fractions continues en caractéristique p

Let K̂ be the field of formal Laurent series in X−1 over the finite field k, and let A be the ring of polynomials in X over k. One of the main results of the paper is to give a natural coding of the (discrete) geodesic flow on the quotient of the Bruhat–Tits tree 𝕋 of PGL2(K̂) by PGL2(A), using the continued fraction expansion of the endpoints of the geodesic lines in 𝕋 (the space of ends of 𝕋 identifies with ℙ 1(K̂). In particular, the invariance of the Haar measure by the Artin transformation can be deduced from the invariance of the Bowen–Margulis measure by the geodesic flow.

Geodesic flow of visibility manifolds

AbstractWe prove that the conservativity of the geodesic flow is equivalent to the ergodicity of the geodesic flow with respect to the Bowen-Margulis measure on visibility manifolds.

Limit sets of discrete groups of isometries of exotic hyperbolic spaces

Let Γ \Gamma be a geometrically finite discrete group of isometries of hyperbolic space H F n \mathcal {H}_{\mathbb {F}}^n , where F = R , C , H \mathbb {F}= \mathbb {R}, \mathbb {C}, \mathbb {H} or O \mathbb {O} (in which case n = 2 n=2 ). We prove that the critical exponent of Γ \Gamma equals the Hausdorff dimension of the limit sets Λ ( Γ ) \Lambda (\Gamma ) and that the smallest eigenvalue of the Laplacian acting on square integrable functions is a quadratic function of either of them (when they are sufficiently large). A generalization of Hopf ergodicity theorem for the geodesic flow with respect to the Bowen-Margulis measure is also proven.