Abstract
Let g be aC 3 negatively curved Riemannian metric on a compact connected orientable surfaceS. LetB be the collection of all metrics resulting from sufficiently small conformal changes of the metricg. (1) Then there is a constantA > 0 such that ifB then the\(\bar d\) distance between the horocycle flowĥ t (Margulis parametrization) of (S, ĝ) and the rescaled horocycle flowh ct of (S, g) is at leastA (∀c > 0). No other dynamical system is known to have such extreme instability. (2) Fix ε > 0. Then there is anN > 0 so that if we are given samples {ξ} N0 {η} N0 which arose from the horocycle flows corresponding to two of the metricsĝ, g ∈B, then either the two samples are\(\bar d\) farther thanA/2 apart or the two surfaces are closer than ε. This holds even if these samples are slightly inaccurate.
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References
W.Ambrose,Representation of ergodic flows, Ann. of Math.42 (1942), 723–739.
D. Anosov,Geodesic flows on closed Riemann manifolds with negative curvature, Proc. Steklov Inst. Math.90 (1967).
R. Bowen,Symbolic dynamics for hyperbolic flows, Am. J. Math.95 (1973), 429–459.
R. Bowen,Equilibrium states and the ergodic theory of Anosov diffeomorphisms, Lecture Notes in Math.470, Springer-Verlag, Berlin, 1975.
R. Bowen and B. Marcus,Unique ergodicity for horocycle foliations, Isr. J. Math.26 (1977), 43–67.
R. Bowen and D. Ruelle,The ergodic theory of axiom A flows, Invent. Math.29 (1975), 181–202.
A. Casson and J. Feldman,A remark about conjugacies of geodesic flows on compact surfaces of negative curvature, unpublished note.
C. Croke,Rigidity for surfaces of nonpositive curvature, Comment. Math. Helv.65 (1989), 150–169.
A. Fathi,Le spectre marqué des longueurs des surfaces sans points conjugués, C.R. Acad. Sci. Paris, Sér. I Math.309 (1989), 621–624.
J. Feldman and D. Ornstein,Semirigidity of horocycle flows over compact surfaces of variable negative curvature, Ergodic Theory and Dynamical Systems7 (1987), 49–72.
G. Hedlund,Fuchsian groups and transitive horocycles, Duke Math. J.2 (1936), 530–542.
M. Hirsch and C. Pugh,Smoothness of horocycle foliations, J. Differ. Geom.10 (1975), 225–238.
M. Hirsch, C. Pugh and M. Shub,Invariant manifolds, Lecture Notes on Math.583, Springer-Verlag, Berlin, 1977.
A. Katok,Four applications of conformal equivalence to geometry and dynamics, Ergodic Theory and Dynamical Systems8 (1988), 139–152.
J. Lehner,A Short Course in Automorphic Functions, Holt, Rinehart and Winston, 1966.
B. Marcus,Unique ergodicity of the horocycle flow: variable negative curvature case, Isr. J. Math.21 (1975), 133–144.
B. Marcus,Ergodic properties of horocycle flows for surfaces of negative curvature, Ann. of Math.105 (1977), 81–105.
G. Margulis,Certain measures associated with U-flows on compact manifolds, Func. Anal. Appl.4 (1970), 55–67.
D. Ornstein,Ergodic Theory, Randomness and Dynamical Systems, Yale University Press, 1974.
D. Ornstein and B. Weiss,How sampling reveals a process, Ann. of Probab.18 (1990), 905–930.
D. Ornstein and B. Weiss,Statistical properties of chaotic systems, Bull. Am. Math. Soc.24 (1991), 11–116.
J-P. Otal,Le spectre marqué des longueurs des surfaces à courbure négative, Ann. of Math.131 (1990), 151–162.
M. Ratner,Rigidity of horocycle flows, Ann. of Math.115 (1982), 587–614.
M. Ratner,Markov partitions for Anosov flows on n-dimensional manifolds, Isr. J. Math.15 (1973), 92–114.
M. Ratner,Invariant measure with respect to an Anosov flow on a three dimensional manifold, Soviet Math. Dokl.10 (1969), 586–588.
C. Robinson,Structural stability of C 1 flows, Lecture Notes in Math.468, Springer-Verlag, Berlin, 1974, pp. 262–277.
V. Rohlin,On the fundamental ideas of measure theory, Am. Math. Soc. Transl.71 (1952).
D. Ruelle,Statistical mechanics of a one-dimensional lattice gas, Comm. Math. Phys.9 (1968), 267–278.
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Troubetzkoy, S.E. Extreme instability of the horocycle flow. J. Anal. Math. 57, 37–63 (1991). https://doi.org/10.1007/BF03041065
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DOI: https://doi.org/10.1007/BF03041065