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Bernoulli flows over maps of the interval. (English) Zbl 0435.58022

MSC:
37C10 Dynamics induced by flows and semiflows
37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior
28D10 One-parameter continuous families of measure-preserving transformations
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[1] D. Anosov,Geodesic flows on closed Riemann manifolds with negative curvature, Proc. Steklov Inst. Math.90 (1967). · Zbl 0176.19101
[2] D. Anosov and Ya. Sinai,Some smooth ergodic systems, Russian Math. Surveys22 (1967), 103–167. · Zbl 0177.42002 · doi:10.1070/RM1967v022n05ABEH001228
[3] R. Bowen,Bernoulli maps of the interval, Israel J. Math.28 (1977), 161–168. · Zbl 0377.28010 · doi:10.1007/BF02759791
[4] R. Bowen,Bernoulli equilibrium states for Axiom A diffeomorphisms, Math. Systems Theory8 (1975), 289–294. · Zbl 0304.28012 · doi:10.1007/BF01780576
[5] R. Bowen,On Axiom A Diffeomorphisms, CBMS Regional Conference at Fargo, Lecture Notes, 1978.
[6] L. Bunimovic,On a class of special flows, Math. USSR Izv.8 (1974), 219–232. · Zbl 0304.58014 · doi:10.1070/IM1974v008n01ABEH002102
[7] B. Gurevič,Some conditions on existence of K-partitions for special flows, Trans. Moscow Math. Soc.
[8] B. Gurevič,The structure of increasing decompositions for special flows. Theor. Probability Appl.10 (1965), 627–645, MR. 35 # 3034. · doi:10.1137/1110077
[9] A. Lasota and G. Yorke,On the existence of invariant measures for piecewise monotomic transformations, Trans. Amer. Math. Soc.186 (1973), 481–488. · Zbl 0298.28015 · doi:10.1090/S0002-9947-1973-0335758-1
[10] E. N. Lorenz,Deterministic nonperiodic flow, J. Atmos. Sci.20 (1963), 130–141. · Zbl 1417.37129 · doi:10.1175/1520-0469(1963)020<0130:DNF>2.0.CO;2
[11] D. Ornstein,Imbedding Bernoulli shifts in flows, inContributions to Ergodic Theory and Probability, Lecture Notes in Math., Springer, Berlin, 1970, pp. 178–218.
[12] D. Ornstein and B. Weiss,Geodesic flows are Bernoullian, Israel J. Math.14 (1973), 184–197. · Zbl 0256.58006 · doi:10.1007/BF02762673
[13] M. Ratner,Anosov flows with Gibbs measures are also Bernoullian, Israel J. Math.17 (1974), 380–391. · Zbl 0304.28011 · doi:10.1007/BF02757140
[14] V. A. Rohlin,New progress in the theory of measure preserving transformations, Russian Math. Surveys15 (1960).
[15] V. A. Rohlin and Ya. G. Sinai,Construction and properties of invariant measurable partitions, Soviet Math. Dokl.2, #6 (1961), 1611–1614. · Zbl 0161.34301
[16] Ya. G. Sinai,Classical dynamical systems with countably multiple Lebesgue spectrum II, Amer. Math. Soc. Transl., Ser. 2,68 (1968), 34–88. · Zbl 0213.34002
[17] R. F. Williams,The structure of Lorenz attractors. · Zbl 0484.58021
[18] S. Wong, Thesis, Berkeley, 1977.
[19] R. L. Adler,f-Expansions Revisited, Springer Lecture Notes318, pp. 1–5.
[20] J. Guckenheimer,A strange, strange attractor, Springer Lecture Notes in Appl. Math. Sci.19 (1976), 368–391.
[21] J. Guckenheimer,Structural stability of the Lorenz attractors. · Zbl 0436.58018
[22] O. Lanford,An Introduction to the Lorenz System, 1976 Duke Turbulence Conference, Duke Univ. Math. Series111 (1977). · Zbl 0357.34027
[23] T. Li and J. A. Yorke,Ergodic transformations from an interval into itself, to appear. · Zbl 0371.28017
[24] W. Parry,On the \(\beta\)-expansion of real numbers, Acta Math. Acad. Sci. Hungar.11 (1960), 401–416. · Zbl 0099.28103 · doi:10.1007/BF02020954
[25] D. Ruelle,Statistical mechanics and dynamical systems, Duke Univ. Math. Series111 (1977). · Zbl 0419.93001
[26] M. Smorodinsky,\(\beta\)-automorphisms are Bernoulli shifts, Acta. Math. Acad. Sci. Hungar.24 (1973), 273–278. · Zbl 0268.28007 · doi:10.1007/BF01958037
[27] P. Walters,Invariant measures and equilibrium states for some mappings which expand distances, to appear. · Zbl 0375.28009
[28] K. Wilkinson,Ergodic properties of a class of piecewise linear transformations, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete31 (1974), 303–328. · Zbl 0299.28015 · doi:10.1007/BF00532869
[29] Ya. G. Sinai,Stochasticity of dynamical systems, to appear. · Zbl 0918.70010
[30] L. Buinmovic and Ya. G. Sinai,Appendix to Sinai’s paper, ”Stochasticity of dynamical systems”.
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