Article Contents
Article Contents

# On uniform convergence in ergodic theorems for a class of skew product transformations

• Consider a class of skew product transformations consisting of an ergodic or a periodic transformation on a probability space $(M, \B,\mu)$ in the base and a semigroup of transformations on another probability space (Ω,$\F,P)$ in the fibre. Under suitable mixing conditions for the fibre transformation, we show that the properties ergodicity, weakly mixing, and strongly mixing are passed on from the base transformation to the skew product (with respect to the product measure). We derive ergodic theorems with respect to the skew product on the product space.
The main aim of this paper is to establish uniform convergence with respect to the base variable for the series of ergodic averages of a function $F$ on $M\times$Ω along the orbits of such a skew product. Assuming a certain growth condition for the coupling function, a strong mixing condition on the fibre transformation, and continuity and integrability conditions for $F,$ we prove uniform convergence in the base and $\L^p(P)$-convergence in the fibre. Under an equicontinuity assumption on $F$ we further show $P$-almost sure convergence in the fibre. Our work has an application in information theory: It implies convergence of the averages of functions on random fields restricted to parts of stair climbing patterns defined by a direction.
Mathematics Subject Classification: Primary: 37A30; Secondary: 37A50.

 Citation:

•  [1] R. Adler and P. Shields, Skew products of Bernoulli shifts with rotations, Israel J. Math., 12 (1972), 215-222.doi: 10.1007/BF02790748. [2] R. Adler and P. Shields, Skew products of Bernoulli shifts with rotations. II, Israel J. Math., 19 (1974), 228-236.doi: 10.1007/BF02757718. [3] H. Anzai, Ergodic skew product transformations on the torus, Osaka Math. J., 3 (1951), 83-99. [4] H. Bauer, Über die Beziehungen einer abstrakten Theorie des Riemann-Integrals zur Theorie Radonscher Maße, Math. Z., 65 (1956), 448-482.doi: 10.1007/BF01473893. [5] A. Bellow and V. Losert, The weighted pointwise ergodic theorem and the individual ergodic theorem along subsequences, Trans. Am. Math. Soc., 288 (1985), 307-345.doi: 10.1090/S0002-9947-1985-0773063-8. [6] J. R. Blum and D. L. Hanson, On the mean ergodic theorem for subsequences, Bull. Am. Math. Soc., 66 (1969), 308-311.doi: 10.1090/S0002-9904-1960-10481-8. [7] J. Brettschneider, Shannon-MacMillan theorems for random fields along curves and lower bounds for surface-order large deviations, Prob. Th. Rel. Fields, 142 (2007), 443-473.doi: 10.1007/s00440-007-0112-z. [8] F. Chersi and A. Volčič, $\lambda$-Equidistributed sequences of partitions and a theorem of the De Bruijn-Post type, Annali di Matematica Pura ed Applicata, 162 (1992), 23-32.doi: 10.1007/BF01759997. [9] N. G. de Bruijn and K. A. Post, A remark on uniformly distributed sequences and Riemann integrability, Indag. Math., 30 (1968), 149-50. [10] J. L. Doob, "Measure Theory," Springer-Verlag, New York, 1993. [11] F. den Hollander and M. Keane, Ergodic properties of color records, Physica A, 138 (1986), 183-193.doi: 10.1016/0378-4371(86)90179-2. [12] N. Friedman, Mixing on sequences, Can. J. Math., 35 (1983), 339-352. [13] H. Furstenberg, Strict ergodicity and transformation of the torus, Amer. J. Math., 83 (1961), 573-601.doi: 10.2307/2372899. [14] H.-O. Georgii, "Gibbs Measures and Phase Transitions," W. de Gruyter, Berlin, 1988. [15] H.-O. Georgii, Mixing properties of induced random transformations, Ergod. Th. and Dynam. Systems, 17 (1997), 839-847.doi: 10.1017/S0143385797086343. [16] O. Hauptmann and C. Pauc, "Differential - und Integralrechnung, Band III," 2. Auflage, Göschen Lehrbücherei, 26, Berlin, 1955. [17] P. Hellekalek and G. Larcher, On the ergodicity of a class of skew products, Israel J. Math., 54 (1986), 301-306.doi: 10.1007/BF02764958. [18] P. Hellekalek and G. Larcher, On Weyl sums and skew products over irrational rotations, Theoret. Comput. Sci., 65 (1989), 189-196.doi: 10.1016/0304-3975(89)90043-1. [19] K. Jacobs, "Measure and Integral," Academic Press, New York, 1978. [20] S. Kakutani, Random ergodic theorems and Markov processes with a stable distribution, in in "Proceedings of the Second Berkeley Symposium on Statistics and Probability" (ed. J. Neyman), Univ. of California Press, Berkeley, (1951), 247-261. [21] A. Katok and B. Hasselblatt, "Introduction to the Modern Theory of Dynamical Systems," Cambridge Univ. Press, Cambridge, 1995. [22] U. Krengel, "Ergodic theorems," W. de Gruyter, Berlin, 1985. [23] M. Lemańczyk and E. Lesigne, Ergodicity of Rokhlin cocycles, J. Anal. Math., 85 (2001), 43-86.doi: 10.1007/BF02788075. [24] L. H. Loomis, Linear functional and content, Amer. J. Math., 76 (1954), 68-82.doi: 10.2307/2372407. [25] I. Meilijson, Mixing properties of a class of skew-products, Israel J. Math., 19 (1974), 266-270.doi: 10.1007/BF02757724. [26] J. Milnor, On the entropy geometry of cellular automata, Complex Syst., 2 (1988), 357-385. [27] I. Oren, Ergodicity of cylinder flows arising from irregularities of distribution, Israel J. Math., 44 (1983), 127-138.doi: 10.1007/BF02760616. [28] D. A. Pask, Skew products over the irrational rotation, Israel J. Math., 69 (1990), 65-74.doi: 10.1007/BF02764730. [29] C. Pauc, Intégrale de partition et intégrale topologique. Familles dérivantes topologiques, C. r. Acad. Sci. Paris, 230 (1950), 810-811. [30] P. Walters, "An Introduction to Ergodic Theory," Springer-Verlag, New York, Heidelberg, Berlin, 1982. [31] H. Weyl, Über die Gleichverteilung von Zahlen mod Eins, Math. Ann., 77 (1915/16), 313-352. doi: 10.1007/BF01475864. [32] Q. Zhang, On skew products of irrational rotations with tori, in "Convergence in Ergodic Theory and Probability" (eds. V. Bergelson, P. March, J. Rosenblatt), Ohio State University Mathematical Research Institute Publications 5, (1996), 435-445.