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Nonuniformly hyperbolic \(K\)-systems are Bernoulli. (English) Zbl 0853.58081

The authors’ abstract: “We prove that those non-uniformly hyperbolic maps and flows (with singularities) that enjoy the \(K\)-property are also Bernoulli. In particular, many billard systems, including those systems of hard balls and stadia that have the \(K\)-property, and hyperbolic billards, such as the Lorentz gas in any dimension, are Bernoulli. We obtain the Bernoulli property for both the billard flows and the associated maps on the boundary of the phase space”.
Reviewer: J.Ombach (Kraków)

MSC:

37D99 Dynamical systems with hyperbolic behavior
37A99 Ergodic theory
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[1] DOI: 10.1007/BF02762673 · Zbl 0256.58006 · doi:10.1007/BF02762673
[2] DOI: 10.1090/S0273-0979-1991-15953-7 · Zbl 0718.58038 · doi:10.1090/S0273-0979-1991-15953-7
[3] DOI: 10.1016/0001-8708(70)90008-3 · Zbl 0227.28014 · doi:10.1016/0001-8708(70)90008-3
[4] DOI: 10.1007/BF02186805 · Zbl 0946.37500 · doi:10.1007/BF02186805
[5] DOI: 10.1007/BF01085490 · Zbl 0748.58015 · doi:10.1007/BF01085490
[6] DOI: 10.1070/RM1991v046n04ABEH002827 · Zbl 0780.58029 · doi:10.1070/RM1991v046n04ABEH002827
[7] DOI: 10.1007/BF01942372 · Zbl 0453.60098 · doi:10.1007/BF01942372
[8] DOI: 10.1007/BF01197884 · Zbl 0421.58017 · doi:10.1007/BF01197884
[9] DOI: 10.1007/BF01205934 · Zbl 0602.58029 · doi:10.1007/BF01205934
[10] DOI: 10.1070/IM1974v008n01ABEH002102 · Zbl 0304.58014 · doi:10.1070/IM1974v008n01ABEH002102
[11] DOI: 10.1016/0378-4371(93)90343-3 · Zbl 0789.58048 · doi:10.1016/0378-4371(93)90343-3
[12] DOI: 10.1007/BF02759791 · Zbl 0377.28010 · doi:10.1007/BF02759791
[13] DOI: 10.1007/BF02099399 · Zbl 0760.58029 · doi:10.1007/BF02099399
[14] DOI: 10.1007/BF01780576 · Zbl 0304.28012 · doi:10.1007/BF01780576
[15] DOI: 10.1007/BF01958037 · Zbl 0268.28007 · doi:10.1007/BF01958037
[16] DOI: 10.1215/S0012-7094-42-00904-9 · Zbl 0063.00065 · doi:10.1215/S0012-7094-42-00904-9
[17] DOI: 10.1070/RM1987v042n03ABEH001421 · Zbl 0644.58007 · doi:10.1070/RM1987v042n03ABEH001421
[18] DOI: 10.1070/RM1970v025n02ABEH003794 · Zbl 0263.58011 · doi:10.1070/RM1970v025n02ABEH003794
[19] DOI: 10.1016/0001-8708(70)90029-0 · Zbl 0197.33502 · doi:10.1016/0001-8708(70)90029-0
[20] DOI: 10.1007/BF01218478 · Zbl 0653.58015 · doi:10.1007/BF01218478
[21] DOI: 10.1007/BF02882235 · Zbl 0284.28007 · doi:10.1007/BF02882235
[22] Ledrappier, Ergod. Th. & Dynam. Sys. 1 pp 77– (1981)
[23] Katok, Invariant Manifolds, Entropy and Billiards: Smooth Maps with Singularities, Springer Lecture Notes in Mathematics 1222 (1986) · doi:10.1007/BFb0099031
[24] Kubo, Nagoya Math. J. 81 pp 1– (1981) · Zbl 0458.58014 · doi:10.1017/S0027763000019127
[25] DOI: 10.1007/BF02771569 · Zbl 0219.28014 · doi:10.1007/BF02771569
[26] DOI: 10.1007/BF01651505 · Zbl 0313.58017 · doi:10.1007/BF01651505
[27] Denker, Dyn. Syst. Ergod. Th., Banach Center Publ 23 (1989)
[28] DOI: 10.1007/BF01200500 · Zbl 0839.60025 · doi:10.1007/BF01200500
[29] DOI: 10.1007/BF02761496 · Zbl 0435.58022 · doi:10.1007/BF02761496
[30] DOI: 10.1007/BF02757140 · Zbl 0304.28011 · doi:10.1007/BF02757140
[31] Pesin, Math. Phys. Rev. 2 pp 53– (1981)
[32] DOI: 10.1070/RM1977v032n04ABEH001639 · Zbl 0383.58011 · doi:10.1070/RM1977v032n04ABEH001639
[33] DOI: 10.1070/IM1977v011n06ABEH001766 · Zbl 0399.58010 · doi:10.1070/IM1977v011n06ABEH001766
[34] Oseledec, Trans. Most: Math. Soc. 19 pp 197– (1968)
[35] DOI: 10.1007/BFb0060654 · doi:10.1007/BFb0060654
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