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A note on the shrinking sector problem for surfaces of variable negative curvature

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Abstract

Given the universal cover Ṽ for a compact surface V of variable negative curvature and a point x̃ 0 ∈ Ṽ, we consider the set of directions \({\widetilde v_0} \in {S_{\widetilde {{x_0}}}}\widetilde V\) for which a narrow sector in the direction ṽ, and chosen to have unit area, contains exactly k points from the orbit of the covering group. We can consider the size of the set of such ṽ in terms of the induced measure on \({S_{{{\widetilde x}_0}}}\widetilde V\) by any Gibbs measure for the geodesic flow. We show that for each k the size of such sets converges as the sector grows narrower and describe these limiting values. The proof involves recasting a similar result by Marklof and Vinogradov, for the particular case of surfaces of constant curvature and the volume measure, by using the strong mixing property for the geodesic flow, relative to the Gibbs measure.

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References

  1. D. V. Anosov, Geodesic Flows on Closed Riemann Manifolds with Negative Curvature (Nauka, Moscow, 1967), Tr. Mat. Inst. im. V.A. Steklova, Akad. Nauk SSSR 90 [Proc. Steklov Inst. Math. 90 (1967)].

    MATH  Google Scholar 

  2. M. Babillot, “On the mixing property for hyperbolic systems,” Isr. J. Math. 129, 61–76 (2002).

    Article  MathSciNet  MATH  Google Scholar 

  3. R. Bowen and D. Ruelle, “The ergodic theory of Axiom A flows,” Invent. Math. 29 (3), 181–202 (1975).

    Article  MathSciNet  MATH  Google Scholar 

  4. N. T. A. Haydn and D. Ruelle, “Equivalence of Gibbs and equilibrium states for homeomorphisms satisfying expansiveness and specification,” Commun. Math. Phys. 148 (1), 155–167 (1992).

    Article  MathSciNet  MATH  Google Scholar 

  5. M. W. Hirsch and C. C. Pugh, “Stable manifolds and hyperbolic sets,” in Global Analysis: Proc. Symp. Pure Math., Berkeley, CA, 1968 (Am. Math. Soc., Providence, RI, 1970), Proc. Symp. Pure Math. 14, pp. 133–163.

    Chapter  Google Scholar 

  6. H. Huber, “Zur analytischen Theorie hyperbolischer Raumformen und Bewegungsgruppen,” Math. Ann. 138, 1–26 (1959).

    Article  MathSciNet  MATH  Google Scholar 

  7. G. A. Margulis, “Applications of ergodic theory to the investigation of manifolds of negative curvature,” Funkts. Anal. Prilozh. 3 (4), 89–90 (1969) [Funct. Anal. Appl. 3, 335–336 (1969)].

    MathSciNet  MATH  Google Scholar 

  8. J. Marklof and I. Vinogradov, “Directions in hyperbolic lattices,” J. Reine Angew. Math., doi: 10.1515/crelle- 2015-0070 (2015).

    Google Scholar 

  9. P. Nicholls, “A lattice point problem in hyperbolic space,” Mich. Math. J. 30 (3), 273–287 (1983).

    Article  MathSciNet  MATH  Google Scholar 

  10. R. Sharp, “Sector estimates for Kleinian groups,” Port. Math. (N.S.) 58 (4), 461–471 (2001).

    MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, University of Warwick, Coventry, CV4 7AL, UK

    Mark Pollicott

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  1. Mark Pollicott
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Correspondence to Mark Pollicott.

Additional information

Published in Russian in Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2017, Vol. 297, pp. 281–291.

In memoriam, Dmitry Victorovich Anosov

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Pollicott, M. A note on the shrinking sector problem for surfaces of variable negative curvature. Proc. Steklov Inst. Math. 297, 254–263 (2017). https://doi.org/10.1134/S0081543817040150

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  • DOI: https://doi.org/10.1134/S0081543817040150

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