Harmonic measures on covers of compact surfaces of nonpositive curvature

Brin, M.; Kifer, Y.

doi:10.1090/S0002-9947-1993-1124163-9

Harmonic measures on covers of compact surfaces of nonpositive curvature
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by M. Brin and Y. Kifer

Trans. Amer. Math. Soc. 340 (1993), 373-393

DOI: https://doi.org/10.1090/S0002-9947-1993-1124163-9

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Abstract:

Let $M$ be the universal cover of a compact nonflat surface $N$ of nonpositive curvature. We show that on the average the Brownian motion on $M$ behaves similarly to the Brownian motion on negatively curved manifolds. We use this to prove that harmonic measures on the sphere at infinity have positive Hausdorff dimension and if the geodesic flow on $N$ is ergodic then the harmonic and geodesic measure classes at infinity are singular unless the curvature is constant.

References

Werner Ballmann, On the Dirichlet problem at infinity for manifolds of nonpositive curvature, Forum Math. 1 (1989), no. 2, 201–213. MR 990144, DOI 10.1515/form.1989.1.201
W. Ballmann and M. Brin, On the ergodicity of geodesic flows, Ergodic Theory Dynam. Systems 2 (1982), no. 3-4, 311–315 (1983). MR 721726, DOI 10.1017/s0143385700001632
Werner Ballmann, Misha Brin, and Patrick Eberlein, Structure of manifolds of nonpositive curvature. I, Ann. of Math. (2) 122 (1985), no. 1, 171–203. MR 799256, DOI 10.2307/1971373
Su Shing Chen and Patrick Eberlein, Isometry groups of simply connected manifolds of nonpositive curvature, Illinois J. Math. 24 (1980), no. 1, 73–103. MR 550653
Isaac Chavel, Eigenvalues in Riemannian geometry, Pure and Applied Mathematics, vol. 115, Academic Press, Inc., Orlando, FL, 1984. Including a chapter by Burton Randol; With an appendix by Jozef Dodziuk. MR 768584
Alfred Gray, Tubes, Addison-Wesley Publishing Company, Advanced Book Program, Redwood City, CA, 1990. MR 1044996
Ernst Heintze and Hans-Christoph Im Hof, Geometry of horospheres, J. Differential Geometry 12 (1977), no. 4, 481–491 (1978). MR 512919
Nobuyuki Ikeda and Shinzo Watanabe, Stochastic differential equations and diffusion processes, 2nd ed., North-Holland Mathematical Library, vol. 24, North-Holland Publishing Co., Amsterdam; Kodansha, Ltd., Tokyo, 1989. MR 1011252
Ioannis Karatzas and Steven E. Shreve, Brownian motion and stochastic calculus, Graduate Texts in Mathematics, vol. 113, Springer-Verlag, New York, 1988. MR 917065, DOI 10.1007/978-1-4684-0302-2
A. Katok, Entropy and closed geodesics, Ergodic Theory Dynam. Systems 2 (1982), no. 3-4, 339–365 (1983). MR 721728, DOI 10.1017/S0143385700001656
Anatole Katok, Four applications of conformal equivalence to geometry and dynamics, Ergodic Theory Dynam. Systems 8$^*$ (1988), no. Charles Conley Memorial Issue, 139–152. MR 967635, DOI 10.1017/S0143385700009391
Y. Kifer, Brownian motion and positive harmonic functions on complete manifolds of nonpositive curvature, From local times to global geometry, control and physics (Coventry, 1984/85) Pitman Res. Notes Math. Ser., vol. 150, Longman Sci. Tech., Harlow, 1986, pp. 187–232. MR 894531
Yuri Kifer and François Ledrappier, Hausdorff dimension of harmonic measures on negatively curved manifolds, Trans. Amer. Math. Soc. 318 (1990), no. 2, 685–704. MR 951889, DOI 10.1090/S0002-9947-1990-0951889-7
François Ledrappier, Propriété de Poisson et courbure négative, C. R. Acad. Sci. Paris Sér. I Math. 305 (1987), no. 5, 191–194 (French, with English summary). MR 903960

Geodesic flows on closed Riemannian manifolds without focal points

11

Mark A. Pinsky, Stochastic Riemannian geometry, Probabilistic analysis and related topics, Vol. 1, Academic Press, New York, 1978, pp. 199–236. MR 0501385
Nicholas Th. Varopoulos, Information theory and harmonic functions, Bull. Sci. Math. (2) 110 (1986), no. 4, 347–389 (English, with French summary). MR 884214