I also wish to express my deep gratitude to Yakov Pesin - my co-author of [PY] - for his generosity in sharing his time and ideas. Anthony Graduate Fellowship for financial support, Professor John Mather for inviting me to visit Princeton University, David DeLatte for his kindness and careful reading of my manuscript, Nantian Qian for his generous assistance, and Kathy Wyland for her excellent typing. However, unlike Odysseus, who had a home for his destiny, there was nothing similar for Virgil's hero, although he had Italy.
On the ideal limit aM of the universal covering M of the negatively curved closed Riemannian manifold M there are three natural measure classes: harmonic measure class {vx}XEM' Lebesgue measure class {mX}xEM' Bowen - Margulis measure class {u x} xEM. A well-known conjecture (A. Katok, F. Ledrappier, D. Sullivan) states that the coincidence of any two of these three measure classes implies that M is locally symmetric. We prove a weaker version of Sullivan's conjecture: horospheres in M have constant mean curvature if and only if mx = Vx for all x EM.
We also show that the function c(x) in Margulis' asymptotic formula c(x) = lim e-hRS(x, R) is almost always. In dimension 2, c( x) is a constant function if and only if the manifold has constant negative curvature.
Introduction
Around the same time as Mostow's early work on stiffness (early 1960s), Russian mathematicians began to systematically study the geodesic flow gt on a closed manifold of negative curvature. If the Anos bifurcation of the geodesic flow gt of M is of class Coo, then gt is Coo-isomorphic to the geodesic flow of a locally symmetric space of negative curvature. In the last section, we reviewed mam's results regarding the stiffness of grids in non-positive curvature manifolds and the geodesic flux stiffness in negative curvature manifolds with a smooth Anos splitting.
We also denote by WSu, WU, W8.9, WS the strong unstable, unstable, strong. stable, stable foliations of the geodesic flow. The harmonic measure v constructed by Ledrappier ([L3]) as the unique equilibrium state of the function . def d.rev) = dt1t=ologk(v(O),v(t),v(oo)). Then the equivalence of two of the three beat classes (the harmonic, Bowen-Marglis, Liouville) implies that M is a locally symmetric space.
The topological entropy of the geodesic arc is equal to the metric entropy hm' if and only if M is locally symmetric. We prove that the stable strong and weak stable leaves of the geodesic flow in M are unique.
Contribution to Sullivan's Conjecture
In the last section, we introduced the harmonic rate v as the unique equilibrium state of the function r. The consequence would easily follow from Proposition 2.1 if the coincidence of the harmonic and the Bowen–Margulis measure implies constant mean curvature of the horospheres. The horospheres of the universal cover X of a compact negatively curved manifold M have constant mean curvature if and only if the harmonic measure Vx coincides at every point with the Lebesgue measure m x.
The topological entropy h of the geodesic flow is equal to the metric entropy hm' if and only if M is locally symmetric. By the minimality of the Poisson kernel, as well as the uniqueness of kernel function (Corollary 5.3 of [AS]), we have. We give an explicit description of the harmonic measure wSS as the weak limit of the normalized spherical measure of geodesic balls.
If m is a mass in M, the mass distributed along the leaves of the leaves D(t)m is defined by. Let {Wt} denote the set of Brownian paths spanning the leaves of the leaves :F (induced by the Riemannian metric on each leaf). By the arguments of Sullivan ([S4]) and Plante ([PI]), for each leaf :F with subexponential growth, the normalized measurements. it is covering weakly to a harmonious mass of leaves.
Since w SU is an invariant measure of the Anosov system and, moreover, it is absolutely continuous along the WIIU foliation. Bowen-Sinai-Margulis measure, we have w8U = m+. A weakly stable foliation of a geodesic arc has a unique harmonic measure wS described by J. A similar result can be given for a unique harmonic measure WU of a weakly stable foliation.
Let WSS be the unique harmonic measure of the stable strong leaf ESS of the geodesic flow. Thus, the harmonic mass is the weak limit of the mean masses on the geodesic spheres. Let RH(v) be the scalar curvature of the horospheres H(v), let Ric(v) be the Ricci curvature of v, then the topological entropy h of the geodesic flow satisfies.
If we denote by K H the Gaussian curvature of H(v) under the induced Riemannian metric, then for any two orthogonal vectors X, Yin T-rr(v)H(v), the Gaussian equation tells us. Take a small open subgroup A of SxM and consider the following transversal T of the leaf Wss.
Integral Formulas of the Laplacian along the Unstable
Brownian Motion on Anosov Foliations,
We assume that both F and the Riemannian metric on its tangent bundle are of class C3• Each leaf L of the leaves inherits a C3 Riemannian structure, making it a connected C3 Riemannian manifold. Thus, each leaf L is completely for diffusion (ie the integral of the heat core over the whole space is equal to one). The leaf diffusion operator D(t) is a continuous affine mapping, and any fixed point will be diffusion invariant in time t. The Markov-Kakutani fixed point theorem ensures that a fixed point exists for all times. I).
Recall that a holonomic invariant measure of the foliation F is a family of measures defined on every transversal of the foliation F, which is invariant under all canonical homeomorphisms of the holonomy pseudogroup (see [PI)). We say that a leaf F has leaves Ck and Riemannian metric Ck on each leaf whose k-lives depend continuously on points in M if for each point in M there is a local parametrization of the leaf F. iii). For any y E V, the Riemannian metric gF drawn on the sheet With WSu (or WSS) we denote the strongly unstable (or strongly stable) foliation of the Anos flow gt or the Anos diffeomorphism f. It is well known that cpu (or cpll) is Holder bounded and has a unique equilibrium state m+ (or m-), which is an invariant ergodic measure of the Anosov system. Since w8U is harmonic and m+ coincides with w8U , we have by Theorem 4.1. for all functions <.p of class C;u. Due to the unique ergodicity of the Wss foliation and the reversibility of Ledrappier's construction, it is easy to see that any family {Tx} xEM of finite measures on 8M satisfying® must coincide with {px} xEM (up to a scalar constant). Let {/-Lx} xEM be the family of Ledrappier-Patterson-Sullivan measures. i) Let X be the geodesic spray. Katok that the normalized measures of the geodesic spheres should converge to the Bowen-Margulis measures J.Lx constructed by Ledrappier ([Ll]). Kaimanovich, An entropy criterion for maximality of the limit of random walks on discrete groups, Soviet Math. Strelcyn, Invariant Manifolds, entropy and billiards; smooth maps with singularities, Reading Notes in Math 1222, Springer-Verlag. Margulis, Applications of ergodic theory to the investigation of sprouts of negative curvature, Punet.
