July  2011, 29(3): 873-891. doi: 10.3934/dcds.2011.29.873

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

1. 

Department of Statistics, University of Warwick, Coventry, CV4 7AL, United Kingdom

Received  October 2009 Revised  August 2010 Published  November 2010

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.
Citation: Julia Brettschneider. On uniform convergence in ergodic theorems for a class of skew product transformations. Discrete & Continuous Dynamical Systems - A, 2011, 29 (3) : 873-891. doi: 10.3934/dcds.2011.29.873
References:
[1]

R. Adler and P. Shields, Skew products of Bernoulli shifts with rotations,, Israel J. Math., 12 (1972), 215. doi: 10.1007/BF02790748. Google Scholar

[2]

R. Adler and P. Shields, Skew products of Bernoulli shifts with rotations. II,, Israel J. Math., 19 (1974), 228. doi: 10.1007/BF02757718. Google Scholar

[3]

H. Anzai, Ergodic skew product transformations on the torus,, Osaka Math. J., 3 (1951), 83. Google Scholar

[4]

H. Bauer, Über die Beziehungen einer abstrakten Theorie des Riemann-Integrals zur Theorie Radonscher Maße,, Math. Z., 65 (1956), 448. doi: 10.1007/BF01473893. Google Scholar

[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. doi: 10.1090/S0002-9947-1985-0773063-8. Google Scholar

[6]

J. R. Blum and D. L. Hanson, On the mean ergodic theorem for subsequences,, Bull. Am. Math. Soc., 66 (1969), 308. doi: 10.1090/S0002-9904-1960-10481-8. Google Scholar

[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. doi: 10.1007/s00440-007-0112-z. Google Scholar

[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. doi: 10.1007/BF01759997. Google Scholar

[9]

N. G. de Bruijn and K. A. Post, A remark on uniformly distributed sequences and Riemann integrability,, Indag. Math., 30 (1968), 149. Google Scholar

[10]

J. L. Doob, "Measure Theory,", Springer-Verlag, (1993). Google Scholar

[11]

F. den Hollander and M. Keane, Ergodic properties of color records,, Physica A, 138 (1986), 183. doi: 10.1016/0378-4371(86)90179-2. Google Scholar

[12]

N. Friedman, Mixing on sequences,, Can. J. Math., 35 (1983), 339. Google Scholar

[13]

H. Furstenberg, Strict ergodicity and transformation of the torus,, Amer. J. Math., 83 (1961), 573. doi: 10.2307/2372899. Google Scholar

[14]

H.-O. Georgii, "Gibbs Measures and Phase Transitions,", W. de Gruyter, (1988). Google Scholar

[15]

H.-O. Georgii, Mixing properties of induced random transformations,, Ergod. Th. and Dynam. Systems, 17 (1997), 839. doi: 10.1017/S0143385797086343. Google Scholar

[16]

O. Hauptmann and C. Pauc, "Differential - und Integralrechnung, Band III," 2. Auflage,, Göschen Lehrbücherei, 26 (1955). Google Scholar

[17]

P. Hellekalek and G. Larcher, On the ergodicity of a class of skew products,, Israel J. Math., 54 (1986), 301. doi: 10.1007/BF02764958. Google Scholar

[18]

P. Hellekalek and G. Larcher, On Weyl sums and skew products over irrational rotations,, Theoret. Comput. Sci., 65 (1989), 189. doi: 10.1016/0304-3975(89)90043-1. Google Scholar

[19]

K. Jacobs, "Measure and Integral,", Academic Press, (1978). Google Scholar

[20]

S. Kakutani, Random ergodic theorems and Markov processes with a stable distribution, in, in, (1951), 247. Google Scholar

[21]

A. Katok and B. Hasselblatt, "Introduction to the Modern Theory of Dynamical Systems,", Cambridge Univ. Press, (1995). Google Scholar

[22]

U. Krengel, "Ergodic theorems,", W. de Gruyter, (1985). Google Scholar

[23]

M. Lemańczyk and E. Lesigne, Ergodicity of Rokhlin cocycles,, J. Anal. Math., 85 (2001), 43. doi: 10.1007/BF02788075. Google Scholar

[24]

L. H. Loomis, Linear functional and content,, Amer. J. Math., 76 (1954), 68. doi: 10.2307/2372407. Google Scholar

[25]

I. Meilijson, Mixing properties of a class of skew-products,, Israel J. Math., 19 (1974), 266. doi: 10.1007/BF02757724. Google Scholar

[26]

J. Milnor, On the entropy geometry of cellular automata,, Complex Syst., 2 (1988), 357. Google Scholar

[27]

I. Oren, Ergodicity of cylinder flows arising from irregularities of distribution,, Israel J. Math., 44 (1983), 127. doi: 10.1007/BF02760616. Google Scholar

[28]

D. A. Pask, Skew products over the irrational rotation,, Israel J. Math., 69 (1990), 65. doi: 10.1007/BF02764730. Google Scholar

[29]

C. Pauc, Intégrale de partition et intégrale topologique. Familles dérivantes topologiques,, C. r. Acad. Sci. Paris, 230 (1950), 810. Google Scholar

[30]

P. Walters, "An Introduction to Ergodic Theory,", Springer-Verlag, (1982). Google Scholar

[31]

H. Weyl, Über die Gleichverteilung von Zahlen mod Eins,, Math. Ann., 77 (): 313. doi: 10.1007/BF01475864. Google Scholar

[32]

Q. Zhang, On skew products of irrational rotations with tori,, in, 5 (1996), 435. Google Scholar

show all references

References:
[1]

R. Adler and P. Shields, Skew products of Bernoulli shifts with rotations,, Israel J. Math., 12 (1972), 215. doi: 10.1007/BF02790748. Google Scholar

[2]

R. Adler and P. Shields, Skew products of Bernoulli shifts with rotations. II,, Israel J. Math., 19 (1974), 228. doi: 10.1007/BF02757718. Google Scholar

[3]

H. Anzai, Ergodic skew product transformations on the torus,, Osaka Math. J., 3 (1951), 83. Google Scholar

[4]

H. Bauer, Über die Beziehungen einer abstrakten Theorie des Riemann-Integrals zur Theorie Radonscher Maße,, Math. Z., 65 (1956), 448. doi: 10.1007/BF01473893. Google Scholar

[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. doi: 10.1090/S0002-9947-1985-0773063-8. Google Scholar

[6]

J. R. Blum and D. L. Hanson, On the mean ergodic theorem for subsequences,, Bull. Am. Math. Soc., 66 (1969), 308. doi: 10.1090/S0002-9904-1960-10481-8. Google Scholar

[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. doi: 10.1007/s00440-007-0112-z. Google Scholar

[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. doi: 10.1007/BF01759997. Google Scholar

[9]

N. G. de Bruijn and K. A. Post, A remark on uniformly distributed sequences and Riemann integrability,, Indag. Math., 30 (1968), 149. Google Scholar

[10]

J. L. Doob, "Measure Theory,", Springer-Verlag, (1993). Google Scholar

[11]

F. den Hollander and M. Keane, Ergodic properties of color records,, Physica A, 138 (1986), 183. doi: 10.1016/0378-4371(86)90179-2. Google Scholar

[12]

N. Friedman, Mixing on sequences,, Can. J. Math., 35 (1983), 339. Google Scholar

[13]

H. Furstenberg, Strict ergodicity and transformation of the torus,, Amer. J. Math., 83 (1961), 573. doi: 10.2307/2372899. Google Scholar

[14]

H.-O. Georgii, "Gibbs Measures and Phase Transitions,", W. de Gruyter, (1988). Google Scholar

[15]

H.-O. Georgii, Mixing properties of induced random transformations,, Ergod. Th. and Dynam. Systems, 17 (1997), 839. doi: 10.1017/S0143385797086343. Google Scholar

[16]

O. Hauptmann and C. Pauc, "Differential - und Integralrechnung, Band III," 2. Auflage,, Göschen Lehrbücherei, 26 (1955). Google Scholar

[17]

P. Hellekalek and G. Larcher, On the ergodicity of a class of skew products,, Israel J. Math., 54 (1986), 301. doi: 10.1007/BF02764958. Google Scholar

[18]

P. Hellekalek and G. Larcher, On Weyl sums and skew products over irrational rotations,, Theoret. Comput. Sci., 65 (1989), 189. doi: 10.1016/0304-3975(89)90043-1. Google Scholar

[19]

K. Jacobs, "Measure and Integral,", Academic Press, (1978). Google Scholar

[20]

S. Kakutani, Random ergodic theorems and Markov processes with a stable distribution, in, in, (1951), 247. Google Scholar

[21]

A. Katok and B. Hasselblatt, "Introduction to the Modern Theory of Dynamical Systems,", Cambridge Univ. Press, (1995). Google Scholar

[22]

U. Krengel, "Ergodic theorems,", W. de Gruyter, (1985). Google Scholar

[23]

M. Lemańczyk and E. Lesigne, Ergodicity of Rokhlin cocycles,, J. Anal. Math., 85 (2001), 43. doi: 10.1007/BF02788075. Google Scholar

[24]

L. H. Loomis, Linear functional and content,, Amer. J. Math., 76 (1954), 68. doi: 10.2307/2372407. Google Scholar

[25]

I. Meilijson, Mixing properties of a class of skew-products,, Israel J. Math., 19 (1974), 266. doi: 10.1007/BF02757724. Google Scholar

[26]

J. Milnor, On the entropy geometry of cellular automata,, Complex Syst., 2 (1988), 357. Google Scholar

[27]

I. Oren, Ergodicity of cylinder flows arising from irregularities of distribution,, Israel J. Math., 44 (1983), 127. doi: 10.1007/BF02760616. Google Scholar

[28]

D. A. Pask, Skew products over the irrational rotation,, Israel J. Math., 69 (1990), 65. doi: 10.1007/BF02764730. Google Scholar

[29]

C. Pauc, Intégrale de partition et intégrale topologique. Familles dérivantes topologiques,, C. r. Acad. Sci. Paris, 230 (1950), 810. Google Scholar

[30]

P. Walters, "An Introduction to Ergodic Theory,", Springer-Verlag, (1982). Google Scholar

[31]

H. Weyl, Über die Gleichverteilung von Zahlen mod Eins,, Math. Ann., 77 (): 313. doi: 10.1007/BF01475864. Google Scholar

[32]

Q. Zhang, On skew products of irrational rotations with tori,, in, 5 (1996), 435. Google Scholar

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