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Harmonic measures on covers of compact surfaces of nonpositive curvature

Authors: M. Brin and Y. Kifer
Journal: Trans. Amer. Math. Soc. 340 (1993), 373-393
MSC: Primary 58G32; Secondary 31B15, 53C21, 60J65
MathSciNet review: 1124163
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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.

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  • [B] W. Ballman, On the Dirichlet problem at infinity for manifolds of nonpositive curvature, Forum Math. 1 (1989), 201-203. MR 990144 (90j:53059)
  • [BB] W. Ballman and M. Brin, On the ergodicity of geodesic flows, Ergodic Theory Dynamical Systems 2 (1982), 311-315. MR 721726 (84k:58174)
  • [BBE] W. Ballman, M. Brin, and P. Eberlein, Structure of manifolds of nonpositive curvature. I, Ann. of Math. (2) 122 (1985), 171-203. MR 799256 (87c:58092a)
  • [CE] S. S. Chen and P. Eberlein, Isometry groups of simply connected manifolds of non-positive curvature, Illinois J. Math. 24 (1980), 73-103. MR 550653 (82k:53052)
  • [Ch] I. Chavel, Eigenvalues in Riemannian geometry, Academic Press, Orlando, 1984. MR 768584 (86g:58140)
  • [G] A. Gray, Tubes, Addison-Wesley, 1989. MR 1044996 (92d:53002)
  • [HI] E. Heintz and H. C. Im Hoff, Geometry of horospheres, J. Differential Geom. 12 (1977), 481-491. MR 512919 (80a:53051)
  • [IW] N. Ikeda and S. Watanabe, Stochastic differential equations and diffusion processes, North-Holland and Kodansha, 1981. MR 1011252 (90m:60069)
  • [KS] I. Karatzas and S. E. Shreve, Brownian motion and stochastic calculus, Graduate Texts in Math., no. 113, Springer-Verlag, New York, 1988. MR 917065 (89c:60096)
  • [Ka1] A. Katok, Entropy and closed geodesics, Ergodic Theory and Dynamical Systems 2 (1982), 339-367. MR 721728 (85b:53047)
  • [Ka2] -, Four applications of conformal equivalence to geometry and dynamics, Ergodic Theory Dynamical Systems 8 (1988), 139-159. MR 967635 (89m:58165)
  • [Ki] Y. Kifer, Brownian motion and positive harmonic functions on complete manifolds of nonpositive curvature, From Local Times to Global Geometry, Control and Physics (K. D. Elworthy, ed.), Pitman Research Notes in Math., no. 150, Longman, Harlow, 1986, pp. 187-232. MR 894531 (88m:58204)
  • [KL] Y. Kifer and F. Ledrappier, Hausdorff dimension of harmonic measures on negatively curved manifolds, Trans. Amer. Math. Soc. 318 (1990), 685-704. MR 951889 (91a:58205)
  • [L] F. Ledrappier, Propriété de Poisson et courbure négative, C. R. Acad. Sci. Paris Ser. I 305 (1987), 191-194. MR 903960 (88m:58205)
  • [P] Ja. B. Pesin, Geodesic flows on closed Riemannian manifolds without focal points, Math. USSR Izv. 11 (1977), 1195-1228.
  • [Pi] M. A. Pinsky, Stochastic Riemannian geometry, Probabilistic Analysis and Related Topics, vol. 1 (A. T. Bharucha-Reid, ed.), Academic Press, New York, 1978, pp. 199-236. MR 0501385 (58:18752)
  • [Va] N. T. Varopoulos, Information theory and harmonic functions, Bull. Sci. Math. (2) 110 (1986), 347-389. MR 884214 (88g:58047)

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Keywords: Harmonic measures, Brownian motion, nonpositive curvature
Article copyright: © Copyright 1993 American Mathematical Society

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