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.

[2]

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

[3]

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

[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.

[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.

[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.

[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.

[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.

[9]

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

[10]

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

[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.

[12]

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

[13]

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

[14]

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

[15]

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

[16]

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

[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.

[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.

[19]

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

[20]

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

[21]

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

[22]

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

[23]

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

[24]

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

[25]

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

[26]

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

[27]

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

[28]

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

[29]

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

[30]

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

[31]

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

[32]

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

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.

[2]

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

[3]

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

[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.

[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.

[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.

[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.

[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.

[9]

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

[10]

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

[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.

[12]

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

[13]

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

[14]

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

[15]

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

[16]

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

[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.

[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.

[19]

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

[20]

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

[21]

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

[22]

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

[23]

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

[24]

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

[25]

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

[26]

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

[27]

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

[28]

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

[29]

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

[30]

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

[31]

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

[32]

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

[1]

Matúš Dirbák. Minimal skew products with hypertransitive or mixing properties. Discrete & Continuous Dynamical Systems - A, 2012, 32 (5) : 1657-1674. doi: 10.3934/dcds.2012.32.1657

[2]

Zuohuan Zheng, Jing Xia, Zhiming Zheng. Necessary and sufficient conditions for semi-uniform ergodic theorems and their applications. Discrete & Continuous Dynamical Systems - A, 2006, 14 (3) : 409-417. doi: 10.3934/dcds.2006.14.409

[3]

Roy Adler, Bruce Kitchens, Michael Shub. Stably ergodic skew products. Discrete & Continuous Dynamical Systems - A, 1996, 2 (3) : 349-350. doi: 10.3934/dcds.1996.2.349

[4]

Jory Griffin, Jens Marklof. Limit theorems for skew translations. Journal of Modern Dynamics, 2014, 8 (2) : 177-189. doi: 10.3934/jmd.2014.8.177

[5]

Roy Adler, Bruce Kitchens, Michael Shub. Errata to "Stably ergodic skew products". Discrete & Continuous Dynamical Systems - A, 1999, 5 (2) : 456-456. doi: 10.3934/dcds.1999.5.456

[6]

Peng Sun. Measures of intermediate entropies for skew product diffeomorphisms. Discrete & Continuous Dynamical Systems - A, 2010, 27 (3) : 1219-1231. doi: 10.3934/dcds.2010.27.1219

[7]

Karma Dajani, Cor Kraaikamp, Pierre Liardet. Ergodic properties of signed binary expansions. Discrete & Continuous Dynamical Systems - A, 2006, 15 (1) : 87-119. doi: 10.3934/dcds.2006.15.87

[8]

Almut Burchard, Gregory R. Chambers, Anne Dranovski. Ergodic properties of folding maps on spheres. Discrete & Continuous Dynamical Systems - A, 2017, 37 (3) : 1183-1200. doi: 10.3934/dcds.2017049

[9]

Gerhard Knieper, Norbert Peyerimhoff. Ergodic properties of isoperimetric domains in spheres. Journal of Modern Dynamics, 2008, 2 (2) : 339-358. doi: 10.3934/jmd.2008.2.339

[10]

Tanja Eisner, Jakub Konieczny. Automatic sequences as good weights for ergodic theorems. Discrete & Continuous Dynamical Systems - A, 2018, 38 (8) : 4087-4115. doi: 10.3934/dcds.2018178

[11]

C.P. Walkden. Stable ergodicity of skew products of one-dimensional hyperbolic flows. Discrete & Continuous Dynamical Systems - A, 1999, 5 (4) : 897-904. doi: 10.3934/dcds.1999.5.897

[12]

François Ledrappier, Omri Sarig. Unique ergodicity for non-uniquely ergodic horocycle flows. Discrete & Continuous Dynamical Systems - A, 2006, 16 (2) : 411-433. doi: 10.3934/dcds.2006.16.411

[13]

P.E. Kloeden, Victor S. Kozyakin. The perturbation of attractors of skew-product flows with a shadowing driving system. Discrete & Continuous Dynamical Systems - A, 2001, 7 (4) : 883-893. doi: 10.3934/dcds.2001.7.883

[14]

Saša Kocić. Reducibility of skew-product systems with multidimensional Brjuno base flows. Discrete & Continuous Dynamical Systems - A, 2011, 29 (1) : 261-283. doi: 10.3934/dcds.2011.29.261

[15]

Tomás Caraballo, Alexandre N. Carvalho, Henrique B. da Costa, José A. Langa. Equi-attraction and continuity of attractors for skew-product semiflows. Discrete & Continuous Dynamical Systems - B, 2016, 21 (9) : 2949-2967. doi: 10.3934/dcdsb.2016081

[16]

Nir Avni. Spectral and mixing properties of actions of amenable groups. Electronic Research Announcements, 2005, 11: 57-63.

[17]

Rafael Alcaraz Barrera. Topological and ergodic properties of symmetric sub-shifts. Discrete & Continuous Dynamical Systems - A, 2014, 34 (11) : 4459-4486. doi: 10.3934/dcds.2014.34.4459

[18]

Mariko Arisawa, Hitoshi Ishii. Some properties of ergodic attractors for controlled dynamical systems. Discrete & Continuous Dynamical Systems - A, 1998, 4 (1) : 43-54. doi: 10.3934/dcds.1998.4.43

[19]

Francesco Cellarosi, Ilya Vinogradov. Ergodic properties of $k$-free integers in number fields. Journal of Modern Dynamics, 2013, 7 (3) : 461-488. doi: 10.3934/jmd.2013.7.461

[20]

Yuri Kifer. Ergodic theorems for nonconventional arrays and an extension of the Szemerédi theorem. Discrete & Continuous Dynamical Systems - A, 2018, 38 (6) : 2687-2716. doi: 10.3934/dcds.2018113

2017 Impact Factor: 1.179

Metrics

  • PDF downloads (6)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]