Acta Mathematica

, Volume 163, Issue 1, pp 1–55 | Cite as

Renewal theorems in symbolic dynamics, with applications to geodesic flows, noneuclidean tessellations and their fractal limits

  • Steven P. Lalley


Symbolic Dynamic Geodesic Flow Fractal Limit Renewal Theorem 
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  1. [1]
    Bowen, R.,Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms. Springer Lecture Notes in Mathematics, 470 (1975).Google Scholar
  2. [2]
    —, Hausdorff dimension of quasicircles.Inst. Hautes Études Sci. Publ. Math., 50 (1979), 11–26.MATHMathSciNetGoogle Scholar
  3. [3]
    Feller, W.,An Introduction to Probability Theory and its Applications, vol. 2. Wiley, New York, 1966.MATHGoogle Scholar
  4. [4]
    Ford, L. R.,Automorphic Functions. McGraw-Hill, New York, 1929.MATHGoogle Scholar
  5. [5]
    Hejhal, D.,The Selberg Trace Formula for PSL(2,R). Springer Lecture Notes in Mathematics, 548 (1976).Google Scholar
  6. [6]
    Huber, H., Über eine neue Klasse automorpher Functionen und eine Gitterpunktproblem in der hyperbolischen Ebene,Comment. Math. Helv., 30 (1956), 20–62.MathSciNetGoogle Scholar
  7. [7]
    Hutchinson, J., Fractals and self similarity.Indiana Univ. Math. J., 30 (1981), 713–747.CrossRefMATHMathSciNetGoogle Scholar
  8. [8]
    Kato, T.,Perturbation Theory for Linear Operators. Springer-Verlag, New York, 1980.MATHGoogle Scholar
  9. [9]
    Kolmogorov, A. N. &Tihomirov, V. M., Epsilon-entropy and epsilon-capacity of sets in functional spaces.Uspekhi Mat. Nauk, 14 (1959), 3–86.MathSciNetMATHGoogle Scholar
  10. [10]
    Krein, M. G., Integral equations on the half-line with a difference kernel.Uspekhi Mat. Nauk, 13 (1958), 3–120.MATHGoogle Scholar
  11. [11]
    Lalley, S., Distribution of periodic orbits of symbolic and Axiom A flows.Adv. in Appl. Math., 8 (1987), 154–193.CrossRefMATHMathSciNetGoogle Scholar
  12. [12]
    —, Regenerative representation for one-dimensional Gibbs states.Ann. Probab., 14 (1986), 1262–1271.MATHMathSciNetGoogle Scholar
  13. [13]
    Lalley, S., Packing and covering functions of some self-similar fractals. Unpublished manuscript (1987).Google Scholar
  14. [14]
    Lax, P. &Phillips, R., The asymptotic distribution of lattice points in euclidean and noneuclidean spaces.J. Funct. Anal., 46 (1982), 280–350.CrossRefMathSciNetMATHGoogle Scholar
  15. [15]
    Lehner, J.,Discontinuous Groups and Automorphic Functions, Amer. Math. Soc., Providence, 1964.MATHGoogle Scholar
  16. [16]
    Mandelbrot, B.,The Fractal Geometry of Nature. Freeman, New York, 1983.Google Scholar
  17. [17]
    Margulis, G., Applications of ergodic theory to the investigation of manifolds of negative curvature.Funktsional. Anal. i Prilozhen, 3 (1969), 89–90.MATHMathSciNetGoogle Scholar
  18. [18]
    Nielsen, J., Untersuchungen zur Topologie der geschlossenen zweiseitigen Flächen.Acta Math., 50 (1927), 189–358.MATHMathSciNetGoogle Scholar
  19. [19]
    Parry, W., Bowen's equidistribution theorem and the Dirichlet density theorem.Ergodic Theory Dynamical Systems, 4 (1984), 171–134.MathSciNetGoogle Scholar
  20. [20]
    Parry, W. &Pollicott, M., An analogue of the prime number theory for closed orbits of Axiom A flows.Ann. of Math., 118 (1983), 573–591.CrossRefMathSciNetGoogle Scholar
  21. [21]
    Patterson, S. J., A lattice point problem in hyperbolic space.Mathematika 22 (1975), 81–88.MATHMathSciNetCrossRefGoogle Scholar
  22. [22]
    —, The limit set of a Fuchsian group.Acta Math., 136 (1976), 241–273.MATHMathSciNetGoogle Scholar
  23. [23]
    Pollicott, M., A complex Ruelle-Perron-Frobenius theorem and two counterexamples.Ergodic Theory Dynamical Systems, 4 (1984), 135–146.MATHMathSciNetGoogle Scholar
  24. [24]
    Rudolph, D., Ergodic behavior of Sullivan's geometric measure on a geometrically finite hyperbolic manifold.Ergodic Theory Dynamical Systems, 2 (1982), 491–512.MATHMathSciNetGoogle Scholar
  25. [25]
    Ruelle, D., Statistical mechanics of a one-dimensional lattice gas.Comm. Math. Phys., 9 (1968), 267–278.CrossRefMATHMathSciNetGoogle Scholar
  26. [26]
    Thermodynamic Formalism. Addison-Wesley, Reading, 1978.MATHGoogle Scholar
  27. [27]
    Series, C., The infinite word problem and limit sets in fuchsian groups.Ergodic Theory Dynamical Systems, 1 (1981), 337–360.MATHMathSciNetGoogle Scholar
  28. [28]
    —, Symbolic dynamics for geodesic flows.Acta Math., 146 (1981), 103–128.MATHMathSciNetGoogle Scholar
  29. [29]
    —, On coding geodesics with continued fractions.Enseign. Math., 29 (1981), 67–76.MATHMathSciNetGoogle Scholar
  30. [30]
    Sullivan, D., On the ergodic theory at infinity of an arbitrary discrete group of hyperbolic motions.Proc. Stony Brook Conf. Kleinian Groups and Reimann Surfaces, Princeton University Press (1978).Google Scholar
  31. [31]
    —, The density at infinity of a discrete group of hyperbolic motions.Inst. Hautes Études Sci. Publ. Math., 50 (1979), 419–450.Google Scholar
  32. [32]
    —, Discrete conformal groups and measurable dynamics.Proc. Sympos. Pure Math., 39 (1983), 169–185.MATHGoogle Scholar
  33. [33]
    —, Entropy, Hausdorff measures new and old, and limit sets of geometrically finite Kleinian groups.Acta Math., 153 (1984), 259–278.MATHMathSciNetGoogle Scholar

Copyright information

© Almqvist & Wiksell 1989

Authors and Affiliations

  • Steven P. Lalley
    • 1
  1. 1.Purdue UniversityWest LafayetteUSA

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