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Ergodic Theory and Dynamical Systems in their Interactions with Arithmetics and Combinatorics pp 327–349Cite as

Weak Mixing for Infinite Measure Invertible Transformations

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Part of the book series: Lecture Notes in Mathematics ((LNM,volume 2213))

Abstract

The weak mixing property for finite measure-preserving transformations has several equivalent characterizations. In infinite measure, many of them turn out to be different. We survey different mixing-like conditions for infinite measure-preserving transformations and discuss interconnectedness.

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Notes

  1. 1.

    While the area of smooth dynamics is an important area for ergodic theory, we will not impose differentiability constraints in this survey. In practical terms, non-differentiable dynamical systems are seeing rapid applications as recurrent neural networks with programmed non-differentiable properties such as rectified linear units and max pooling. Also, neurological spiking is naturally modeled as a discontinuous process.

  2. 2.

    Strong mixing is often referred to as mixing.

References

  1. J. Aaronson, Rational ergodicity and a metric invariant for Markov shifts. Israel J. Math. 27(2), 93–123 (1977)

    Article  MathSciNet  Google Scholar 

  2. J. Aaronson, Rational ergodicity bounded rational ergodicity and some continuous measures on the circle. Israel J. Math. 33(3–4), 181–197 (1979). https://doi.org/10.1007/BF02762160.

    Article  MathSciNet  Google Scholar 

  3. J. Aaronson, An Introduction to Infinite Ergodic Theory. Mathematical Surveys and Monographs, vol. 50 (American Mathematical Society, Providence, RI, 1997)

    Google Scholar 

  4. J. Aaronson, Rational weak mixing in infinite measure spaces. Ergod. Theory Dyn. Syst. 33(6), 1611–1643 (2013). https://doi.org/10.1017/etds.2012.102

    Article  MathSciNet  Google Scholar 

  5. J. Aaronson, Conditions for rational weak mixing. Stoch. Dyn. 16(2), 1660004, 12 (2016). https://doi.org/10.1142/S0219493716600042

    Article  MathSciNet  Google Scholar 

  6. J. Aaronson, M. Nadkarni, L ∞ eigenvalues and L2 spectra of nonsingular transformations. Proc. Lond. Math. Soc. (3) 55(3), 538–570 (1987). https://doi.org/10.1112/plms/s3-55.3.538

  7. J. Aaronson, H. Nakada, Multiple recurrence of Markov shifts and other infinite measure preserving transformations. Israel J. Math. 117, 285–310 (2000). https://doi.org/10.1007/BF02773574

    Article  MathSciNet  Google Scholar 

  8. J. Aaronson, B. Weiss, Symmetric Birkhoff sums in infinite ergodic theory (2013), http://arxiv.org/abs/1307.7490

  9. J. Aaronson, M. Lin, B. Weiss, Mixing properties of Markov operators and ergodic transformations, and ergodicity of Cartesian products. Israel J. Math. 33(3–4), 198–224 (1979). https://doi.org/10.1007/BF02762161

    Article  MathSciNet  Google Scholar 

  10. C. Abraham, G. Biau, B. Cadre, Chaotic properties of mappings on a probability space. J. Math. Anal. Appl. 266(2), 420–431 (2002). https://doi.org/10.1006/jmaa.2001.7754

    Article  MathSciNet  Google Scholar 

  11. T.M. Adams, Smorodinsky’s conjecture on rank-one mixing. Proc. Am. Math. Soc. 126(3), 739–744 (1998). https://doi.org/10.1090/S0002-9939-98-04082-9

  12. T. Adams, Rigidity sequences of power rationally weakly mixing transformations (2015), https://arxiv.org/abs/1503.05806

  13. T. Adams, C.E. Silva, Rank-one power weakly mixing non-singular transformations. Ergod. Theory Dyn. Syst. 21(5), 1321–1332 (2001). https://doi.org/10.1017/S0143385701001626

    Article  MathSciNet  Google Scholar 

  14. T. Adams, C.E. Silva, On infinite transformations with maximal control of ergodic two-fold product powers. Israel J. Math. 209(2), 929–948 (2015). https://doi.org/10.1007/s11856-015-1241-1

    Article  MathSciNet  Google Scholar 

  15. T. Adams, C.E. Silva, Weak rational ergodicity does not imply rational ergodicity. Israel J. Math. 214(1), 491–506 (2016). https://doi.org/10.1007/s11856-016-1371-0

    Article  MathSciNet  Google Scholar 

  16. T. Adams, N. Friedman, C.E. Silva, Rank-one weak mixing for non-singular transformations. Israel J. Math. 102, 269–281 (1997). https://doi.org/10.1007/BF02773802

    Article  MathSciNet  Google Scholar 

  17. O.N. Ageev, C.E. Silva, Genericity of rigid and multiply recurrent innite measure-preserving and nonsingular transformations, in Proceedings of the 16th Summer Conference on General Topology and its Applications (New York), vol. 26, pp. 357–365 (2001/2002)

    Google Scholar 

  18. D. Aiello, H. Diao, Z. Fan, D.O. King, J. Lin, C.E. Silva, Measurable time-restricted sensitivity. Nonlinearity 25(12), 3313–3325 (2012). https://doi.org/10.1088/0951-7715/25/12/3313

    Article  MathSciNet  Google Scholar 

  19. J. Auslander, J.A. Yorke, Interval maps, factors of maps, and chaos. Tôhoku Math. J. (2) 32(2), 177–188 (1980). https://doi.org/10.2748/tmj/1178229634

    Article  MathSciNet  Google Scholar 

  20. J.R. Baxter, A class of ergodic transformations having simple spectrum. Proc. Am. Math. Soc. 27, 275–279 (1971). https://doi.org/10.2307/2036306

    Article  MathSciNet  Google Scholar 

  21. R.L. Bayless, K.B. Yancey, Weakly mixing and rigid rank-one transformations preserving an infinite measure. New York J. Math. 21, 615–636 (2015)

    MathSciNet  MATH  Google Scholar 

  22. V. Bergelson, A. Gorodnik, Weakly Mixing Group Actions: A Brief Survey and an Example. Modern Dynamical Systems and Applications (Cambridge University Press, Cambridge, 2004), pp. 3–25

    MATH  Google Scholar 

  23. V. Bergelson, A. del Junco, M. Lemańczyk, J. Rosenblatt, Rigidity and non-recurrence along sequences. Ergod. Theory Dyn. Syst. 34(5), 1464–1502 (2014). https://doi.org/10.1017/etds.2013.5

    Article  MathSciNet  Google Scholar 

  24. A. Bowles, L. Fidkowski, A.E. Marinello, C.E. Silva, Double ergodicity of nonsingular transformations and infinite measure-preserving staircase transformations. Ill. J. Math. 45(3), 999–1019 (2001)

    MathSciNet  MATH  Google Scholar 

  25. F. Bozgan, A. Sanchez, C.E. Silva, D. Stevens, J. Wang, Sub-sequence bounded rational ergodicity of rank-one transformations. Dyn. Syst. 30(1), 70–84 (2015). https://doi.org/10.1080/14689367.2014.970518

    Article  MathSciNet  Google Scholar 

  26. B. Cadre, P. Jacob, On pairwise sensitivity. J. Math. Anal. Appl. 309(1), 375–382 (2005). https://doi.org/10.1016/j.jmaa.2005.01.061

    Article  MathSciNet  Google Scholar 

  27. J.R. Choksi, S. Kakutani, Residuality of ergodic measurable transformations and of ergodic transformations which preserve an infinite measure. Indiana Univ. Math. J. 28(3), 453–469 (1979). https://doi.org/10.1512/iumj.1979.28.28032

    Article  MathSciNet  Google Scholar 

  28. J. Clancy, R. Friedberg, I. Kasmalkar, I. Loh, T. Pădurariu, C.E. Silva, S. Vasudevan, Ergodicity and conservativity of products of infinite transformations and their inverses. Colloq. Math. 143 (2), 271–291 (2016)

    MathSciNet  MATH  Google Scholar 

  29. I. Dai, X. Garcia, T. Pădurariu, C.E. Silva, On rationally ergodic and rationally weakly mixing rank-one transformations. Ergod. Theory Dyn. Syst. 35 (4), 1141–1164 (2015). https://doi.org/10.1017/etds.2013.96

    Article  MathSciNet  Google Scholar 

  30. A.I. Danilenko, Funny rank-one weak mixing for nonsingular abelian actions. Israel J. Math. 121, 29–54 (2001). https://doi.org/10.1007/BF02802494

    Article  MathSciNet  Google Scholar 

  31. A.I. Danilenko, Infinite rank one actions and nonsingular Chacon transformations. Ill. J. Math. 48(3), 769–786 (2004)

    MathSciNet  MATH  Google Scholar 

  32. A.I. Danilenko, (C, F)-Actions in Ergodic Theory. Geometry and Dynamics of Groups and Spaces (Springer Science and Business Media, Berlin, 2008), pp. 325–351

    Google Scholar 

  33. A.I. Danilenko, Finite ergodic index and asymmetry for infinite measure preserving actions. Proc. Am. Math. Soc. 144(6), 2521–2532 (2016). https://doi.org/10.1090/proc/12906

    Article  MathSciNet  Google Scholar 

  34. A.I. Danilenko, Directional recurrence and directional rigidity for infinite measure preserving actions of nilpotent lattices. Ergod. Theory Dyn. Syst. 37 (6), 1841–1861 (2017). https://doi.org/10.1017/etds.2015.127

    Article  MathSciNet  Google Scholar 

  35. A.I. Danilenko, V.V. Ryzhikov, Mixing constructions with infinite invariant measure and spectral multiplicities. Ergod. Theory Dyn. Syst. 31 (3), 853–873 (2011). https://doi.org/10.1017/S0143385710000052

    Article  MathSciNet  Google Scholar 

  36. A.I. Danilenko, C.E. Silva, Multiple and polynomial recurrence for abelian actions in infinite measure. J. Lond. Math. Soc. (2) 69 (1), 183–200 (2004). https://doi.org/10.1112/S0024610703004885

    Article  MathSciNet  Google Scholar 

  37. A.I. Danilenko, C.E. Silva, Ergodic Theory: Non-Singular Transformations (Springer, New York, 2012)

    Google Scholar 

  38. S.L. Day, B.R. Grivna, E.P. McCartney, C.E. Silva, Power weakly mixing infinite transformations. New York J. Math. 5, 17–24 (1999) (electronic)

    Google Scholar 

  39. R.L. Devaney, An Introduction to Chaotic Dynamical Systems. Addison-Wesley Studies in Nonlinearity, 2nd edn. (Addison-Wesley Publishing Company Advanced Book Program, Redwood City, CA, 1989)

    Google Scholar 

  40. Y.N. Dowker, P. Erdős, Some examples in ergodic theory. Proc. Lond. Math. Soc. (3) 9, 227–241 (1959). https://doi.org/10.1112/plms/s3-9.2.227

    Article  MathSciNet  Google Scholar 

  41. H.A. Dye, On the ergodic mixing theorem. Trans. Am. Math. Soc. 118, 123–130 (1965). https://doi.org/10.2307/1993948

    Article  MathSciNet  Google Scholar 

  42. S.J. Eigen, Ergodic cartesian products à la triangle sets, in Measure Theory Oberwolfach 1983 (Oberwolfach, 1983) (1984), pp. 263–270. https://doi.org/10.1007/BFb0072620

    Google Scholar 

  43. S. Eigen, A. Hajian, K. Halverson, Multiple recurrence and infinite measure preserving odometers. Israel J. Math. 108, 37–44 (1998). https://doi.org/10.1007/BF02783041

    Article  MathSciNet  Google Scholar 

  44. S. Eigen, A. Hajian, Y. Ito, V. Prasad, Weakly Wandering Sequences in Ergodic Theory. Springer Monographs in Mathematics (Springer, Tokyo, 2014)

    Google Scholar 

  45. S. Ferenczi, Systems of finite rank. Colloq. Math. 73 (1), 35–65 (1997)

    Article  MathSciNet  Google Scholar 

  46. N.A. Friedman, Introduction to Ergodic Theory. Van Nostrand Reinhold Mathematical Studies, No. 29 (Van Nostrand Reinhold Co., New York, 1970)

    Google Scholar 

  47. N.A. Friedman, Mixing Transformations in an Infinite Measure Space. Studies in Probability and Ergodic Theory (Academic Press, New York, 1978), pp. 167–184

    Google Scholar 

  48. N.A. Friedman, J.L. King, Rank one lightly mixing. Israel J. Math. 73 (3), 281–288 (1991). https://doi.org/10.1007/BF02773841

    Article  MathSciNet  Google Scholar 

  49. N.A. Friedman, D.S. Ornstein, On partially mixing transformations. Indiana Univ. Math. J. 20, 767–775 (1970/1971). https://doi.org/10.1512/iumj.1971.20.20061

    Article  MathSciNet  Google Scholar 

  50. H. Furstenberg, Recurrence in Ergodic Theory and Combinatorial Number Theory (Princeton University Press, Princeton, 1981). M. B. Porter Lectures

    Google Scholar 

  51. H. Furstenberg, B. Weiss, The finite multipliers of infinite ergodic transformations, in The Structure of Attractors in Dynamical Systems (Proc. Conf., North Dakota State Univ., Fargo, ND,1977) (1978), pp. 127–132

    Chapter  Google Scholar 

  52. E. Glasner, B. Weiss, Sensitive dependence on initial conditions. Nonlinearity 6 (6), 1067–1075 (1993)

    Article  MathSciNet  Google Scholar 

  53. E. Glasner, B. Weiss, Weak mixing properties for non-singular actions. Ergod. Theory Dyn. Syst. 36 (7), 2203–2217 (2016). https://doi.org/10.1017/etds.2015.16

    Article  MathSciNet  Google Scholar 

  54. A. Glücksam, Ergodic multiplier properties. Ergod. Theory Dyn. Syst. 36 (3), 794–815 (2016). https://doi.org/10.1017/etds.2014.79

    Article  MathSciNet  Google Scholar 

  55. I. Grigoriev, M.C. Iordan, A. Lubin, N. Ince, C.E. Silva, On μ-compatible metrics and measurable sensitivity. Colloq. Math. 126 (1), 53–72 (2012). https://doi.org/10.4064/cm126-1-3

    Article  MathSciNet  Google Scholar 

  56. K. Gruher, F. Hines, D. Patel, C.E. Silva, R. Waelder, Power weak mixing does not imply multiple recurrence in infinite measure and other counterexamples. New York J. Math. 9, 1–22 (2003) (electronic)

    Google Scholar 

  57. J. Guckenheimer, Sensitive dependence to initial conditions for one-dimensional maps. Comm. Math. Phys. 70(2), 133–160 (1979)

    Article  MathSciNet  Google Scholar 

  58. A.B. Hajian, S. Kakutani, Weakly wandering sets and invariant measures. Trans. Am. Math. Soc. 110, 136–151 (1964)

    Article  MathSciNet  Google Scholar 

  59. A.B. Hajian, S. Kakutani, Example of an ergodic measure preserving transformation on an infinite measure space, in Contributions to Ergodic Theory and Probability (Proc. Conf., Ohio State Univ., Columbus, OH, 1970) (1970), pp. 45–52

    Google Scholar 

  60. J. Hallett, L. Manuelli, C.E. Silva, On Li-Yorke measurable sensitivity. Proc. Am. Math. Soc. 143(6), 2411–2426 (2015). https://doi.org/10.1090/S0002-9939-2015-12430-6

    Article  MathSciNet  Google Scholar 

  61. P.R. Halmos, Approximation theories for measure preserving transformations. Trans. Am. Math. Soc. 55, 1–18 (1944). https://doi.org/10.2307/1990137

    Article  MathSciNet  Google Scholar 

  62. P.R. Halmos, In general a measure preserving transformation is mixing. Ann. Math (2) 45, 786–792 (1944). https://doi.org/10.2307/1969304

  63. P.R. Halmos, Invariant measures. Ann. of Math.(2) 48, 735–754 (1947). https://doi.org/10.2307/1969138

    Article  MathSciNet  Google Scholar 

  64. P.R. Halmos, Lectures on Ergodic Theory (Chelsea Publishing Co., New York, 1960)

    MATH  Google Scholar 

  65. E. Hopf, Ergodentheorie. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3, 5 (Springer, Berlin, 1937)

    Google Scholar 

  66. W. Huang, P. Lu, X. Ye, Measure-theoretical sensitivity and equicontinuity. Israel J. Math. 183, 233–283 (2011). https://doi.org/10.1007/s11856-011-0049-x

    Article  MathSciNet  Google Scholar 

  67. J. James, T. Koberda, K. Lindsey, C.E. Silva, P. Speh, Measurable sensitivity. Proc. Am. Math. Soc. 136(10), 3549–3559 (2008). https://doi.org/10.1090/S0002-9939-08-09294-0

    Article  MathSciNet  Google Scholar 

  68. S. Kakutani, W. Parry, Infinite measure preserving transformations with “mixing”. Bull. Am. Math. Soc. 69, 752–756 (1963)

    Article  MathSciNet  Google Scholar 

  69. J.L. King, Joining-rank and the structure of finite rank mixing transformations. J. Anal. Math. 51, 182–227 (1988). https://doi.org/10.1007/BF02791123

    Article  MathSciNet  Google Scholar 

  70. U. Krengel, Entropy of conservative transformations. Z. Wahrscheinlichkeitstheorie und Verw Gebiete 7, 161–181 (1967). https://doi.org/10.1007/BF00532635

    Article  MathSciNet  Google Scholar 

  71. U. Krengel, L. Sucheston, On mixing in infinite measure spaces. Z. Wahrscheinlichkeit-stheorie und Verw Gebiete 13, 150–164 (1969). https://doi.org/10.1007/BF00537021

    Article  MathSciNet  Google Scholar 

  72. K. Krickeberg, Strong mixing properties of Markov chains with infinite invariant measure, in Proceedings of Fifth Berkeley Symposium on Mathematical Statistics and Probability (Berkeley, CA, 1965/66), Vol. II: Contributions to Probability Theory Part 2 (1967), pp. 431–446

    Google Scholar 

  73. M. Lenci, On infinite-volume mixing. Comm. Math. Phys. 298(2), 485–514 (2010). https://doi.org/10.1007/s00220-010-1043-6

    Article  MathSciNet  Google Scholar 

  74. I. Loh, C.E. Silva, Strict doubly ergodic infinite transformations. Dyn. Syst. 1–25 (2017, to appear). https://doi.org/10.1080/14689367.2017.1280665

  75. I. Loh, C.E. Silva, B. Athiwaratkun, Infinite symmetric ergodic index and related examples in infinite measure. Stud. Math. (to appear). https://arxiv.org/abs/1702.01455 Isaac Loh, Cesar E. Silva, and Ben Athiwaratkun, Innite symmetric ergodic index and related examples in innite measure, Studia Mathematica, to appear.

  76. C.A. Morales, Partition sensitivity for measurable maps. Math. Bohem. 138(2), 133–148 (2013)

    MathSciNet  MATH  Google Scholar 

  77. E.J. Muehlegger, A.S. Raich, C.E. Silva, M.P. Touloumtzis, B. Narasimhan, W. Zhao, Infinite ergodic index Z d-actions in infinite measure. Colloq. Math. 82(2), 167–190 (1999)

    Article  MathSciNet  Google Scholar 

  78. D.S. Ornstein, On the Root Problem in Ergodic Theory (University California Press, Berkeley, 1972)

    MATH  Google Scholar 

  79. W. Parry, Ergodic and spectral analysis of certain infinite measure preserving transformations. Proc. Am. Math. Soc. 16, 960–966 (1965)

    Article  MathSciNet  Google Scholar 

  80. W. Parry, Topics in Ergodic Theory. Cambridge Tracts in Mathematics, vol. 75 (Cambridge University Press, Cambridge, 2004). Reprint of the 1981 original

    Google Scholar 

  81. K.E. Petersen, Disjointness and weak mixing of minimal sets. Proc. Am. Math. Soc. 24, 278–280 (1970). https://doi.org/10.2307/2036347

    Article  MathSciNet  Google Scholar 

  82. V. Rohlin, A “general” measure-preserving transformation is not mixing. Doklady Akad. Nauk SSSR (N.S.) 60, 349–351 (1948)

    Google Scholar 

  83. D.J. Rudolph, Fundamentals of Measurable Dynamics. Ergodic Theory on Lebesgue Spaces (Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1990)

    Google Scholar 

  84. D.J. Rudolph, C.E. Silva, Minimal self-joinings for nonsingular transformations. Ergod. Theory Dyn. Syst. 9(4), 759–800 (1989). https://doi.org/10.1017/S0143385700005320

    Article  MathSciNet  Google Scholar 

  85. D. Ruelle, Dynamical systems with turbulent behavior, in Mathematical Problems in Theoretical Physics (Proc. Internat. Conf., Univ. Rome, Rome, 1977) (1978), pp. 341–360

    Google Scholar 

  86. U. Sachdeva, On category of mixing in infinite measure spaces. Math. Syst. Theory 5, 319–330 (1971)

    Article  MathSciNet  Google Scholar 

  87. C.E. Silva, Invitation to Ergodic Theory. Student Mathematical Library, vol. 42 (American Mathematical Society, Providence, 2008)

    Google Scholar 

  88. C.E. Silva, On Mixing-Like Notions in Infinite Measure. Am. Math. Mon. 124(9), 807–825 (2017)

    Article  MathSciNet  Google Scholar 

  89. C.E. Silva, P. Thieullen, A skew product entropy for nonsingular transformations. J. Lond. Math. Soc. (2) 52(3), 497–516 (1995)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

We would like to thank the referee and Isaac Loh for suggestions and corrections.

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Authors and Affiliations

  1. U.S. Government, Ft. Meade, MD, USA

    Terrence Adams

  2. Department of Mathematics, Williams College, Williamstown, MA, USA

    Cesar E. Silva

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  1. Terrence Adams
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Editors and Affiliations

  1. CNRS UMR 7373, Institut de Mathématiques de Marseille, Marseille, France

    Sébastien Ferenczi

  2. Faculty of Mathematics and Computer Science, Nicolaus Copernicus University, Toruń, Poland

    Joanna Kułaga-Przymus

  3. Faculty of Mathematics and Computer Science, Nicolaus Copernicus University, Toruń, Poland

    Mariusz Lemańczyk

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Adams, T., Silva, C.E. (2018). Weak Mixing for Infinite Measure Invertible Transformations. In: Ferenczi, S., Kułaga-Przymus, J., Lemańczyk, M. (eds) Ergodic Theory and Dynamical Systems in their Interactions with Arithmetics and Combinatorics. Lecture Notes in Mathematics, vol 2213. Springer, Cham. https://doi.org/10.1007/978-3-319-74908-2_17

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