Geodesic
Criteria of divergence almost everywhere in ergodic theory
Zapiski Nauchnykh Seminarov POMI, Probability and statistics. Part 22, Tome 441 (2015), pp. 73-118 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice du chapitre de livre

In this expository paper, we survey nowadays classical tools or criteria used in problems of convergence everywhere to build counterexamples: the Stein continuity principle, Bourgain's entropy criteria and Kakutani–Rochlin lemma, most classical device for these questions in ergodic theory. First, we state a $L^1$-version of the continuity principle and give an example of its usefulness by applying it to some famous problem on divergence almost everywhere of Fourier series. Next we particularly focus on entropy criteria in $L^p$, $2\le p\le\infty$, and provide detailed proofs. We also study the link between the associated maximal operators and the canonical Gaussian process on $L^2$. We further study the corresponding criterion in $L^p$, $1, using properties of $p$-stable processes. Finally we consider Kakutani–Rochlin's lemma, one of the most frequently used tool in ergodic theory, by stating and proving a criterion for a.e. divergence of weighted ergodic averages. 
@article{ZNSL_2015_441_a6,
     author = {M. J. G. Weber},
     title = {Criteria of divergence almost everywhere in ergodic theory},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {73--118},
     year = {2015},
     volume = {441},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2015_441_a6/}
}
TY  - JOUR
AU  - M. J. G. Weber
TI  - Criteria of divergence almost everywhere in ergodic theory
JO  - Zapiski Nauchnykh Seminarov POMI
PY  - 2015
SP  - 73
EP  - 118
VL  - 441
UR  - http://geodesic.mathdoc.fr/item/ZNSL_2015_441_a6/
LA  - en
ID  - ZNSL_2015_441_a6
ER  - 
%0 Journal Article
%A M. J. G. Weber
%T Criteria of divergence almost everywhere in ergodic theory
%J Zapiski Nauchnykh Seminarov POMI
%D 2015
%P 73-118
%V 441
%U http://geodesic.mathdoc.fr/item/ZNSL_2015_441_a6/
%G en
%F ZNSL_2015_441_a6
M. J. G. Weber. Criteria of divergence almost everywhere in ergodic theory. Zapiski Nauchnykh Seminarov POMI, Probability and statistics. Part 22, Tome 441 (2015), pp. 73-118. http://geodesic.mathdoc.fr/item/ZNSL_2015_441_a6/

[1] J. Bourgain, “Problems of almost everywhere convergence related to harmonic analysis and number theory”, Israel J. Math., 71 (1990), 97–127 | DOI | MR | Zbl

[2] J. Bourgain, “Almost sure convergence and bounded entropy”, Israel J. Math., 63 (1988), 79–95 | DOI | MR

[3] J. Bourgain, “An approach to pointwise ergodic theorems”, Geometric aspects of functional analysis (1986/87), Lecture Notes in Math., 1317, Springer, Berlin, 1988, 204–223 | DOI | MR

[4] A. Bellow, R. Jones, “A Banach principle for $L^\infty$”, Adv. Math., 120 (1996), 155–172 | DOI | MR | Zbl

[5] D. L. Burkholder, “Maximal inequalities as necessary conditions for almost everywhere convergence”, Z. Wahrscheinlichkeitsth. Verw. Geb., 3 (1964), 75–88 | DOI | MR | Zbl

[6] I. Berkes, M. Weber, “On series $\sum c_k f(kx)$ and Khinchin's conjecture”, Israel J. Math., 201:2 (2014), 593–609 | DOI | MR | Zbl

[7] Y. Deniel, “On the a.s. Cesàro-$\alpha$ convergence for stationary of orthogonal sequences”, J. Theoret. Probab., 2 (1989), 475–485 | DOI | MR | Zbl

[8] R. M. Dudley, “Sample functions of the Gaussian process”, Ann. Probab., 1:1 (1973), 66–103 | DOI | MR | Zbl

[9] R. M. Dudley, “The size of compact subsets of Hilbert space and continuity of Gaussian processes”, J. Funct. Anal., 1 (1967), 290–330 | DOI | MR | Zbl

[10] A. Garsia, Topics in Almost Everywhere Convergence, Markham Publ. Co., Chicago, 1970 | MR | Zbl

[11] A. Ya. Helemskiĭ, G. M. Henkin, “Embeddings of compacta into ellipsoids”, Vestnik Moskov. Univ. Ser. I. Matem. Mekhan., 1963, no. 2, 3–12 (in Russian) | MR

[12] A. N. Kolmogorov, “Asymptotic characteristics of some completely bounded metric spaces”, Dokl. Akad. Nauk SSSR, 108 (1956), 585–589 (in Russian) | MR

[13] A. N. Kolmogorov, “Sur les fonctions harmoniques conjuguées et les séries de Fourier”, Fund. Math., 7 (1925), 23–28

[14] A. N. Kolmogorov, “Une série de Fourier–Lebesgue divergente presque partout”, Fund. Math., 4 (1923), 324–328

[15] A. N. Kolmogorov, “Une série de Fourier–Lebesgue divergente presque partout”, C. R. Acad. Sci. Paris Sér. I, Math., 183 (1926), 1327–1329

[16] Amer. Math. Soc. Transl., 17 (1961), 277–364 | MR | MR | Zbl | Zbl

[17] M. T. Lacey, “Carleson's theorem, proof, complements, variations”, Publ. Math., 48 (2004), 251–307 | DOI | MR | Zbl

[18] M. T. Lacey, “The return time theorem fails on infinite measure preserving systems”, Ann. Inst. Henri Poincaré, 33:4 (1997), 491–495 | DOI | MR | Zbl

[19] J. Lamperti, “On the isometries of certain function-spaces”, Pacific J. Math., 8 (1958), 459–466 | DOI | MR | Zbl

[20] E. Lesigne, “On the sequence of integer parts of a good sequence for the ergodic theorem”, Comment. Math. Univ. Carolin., 36 (1995), 737–743 | MR | Zbl

[21] M. Lifshits, M. Weber, “Oscillations of the Gaussian Stein's elements”, High dimensional probability (Oberwolfach, 1996), Progr. Probab., 43, Birkhaüser, Basel, 1998, 249–261 | MR | Zbl

[22] M. Lifshits, M. Weber, “Tightness of stochastic families arising from randomization procedure”, Asymptotic methods in probability and statistics with applications (St. Petersburg, 1998), Stat. Ind. Technol., Birkhäuser, Boston, 2001, 143–158 | MR

[23] G. G. Lorentz, “Metric entropy and approximation”, Bull. Amer. Math. Soc., 72 (1966), 903–937 | DOI | MR | Zbl

[24] M. Marcus, G. Pisier, “Characterizations of almost surely $p$-stable random Fourier series and strongly continuous stationary processes”, Acta Math., 152 (1984), 245–301 | DOI | MR | Zbl

[25] R. A. Raimi, “Compact transformations and the $k$-topology in Hilbert space”, Proc. Amer. Math. Soc., 6 (1955), 643–646 | MR | Zbl

[26] J. M. Rosenblatt, M. Wierdl, “Pointwise Ergodic Theorems via Harmonic Analysis”, Proceedings of the Conference on Ergodic Theory and its Connections with Harmonic Analysis, Cambridge University Press, Alexandria, Egypt, 1994, 3–151 | MR

[27] S. Sawyer, “Maximal inequalities of weak type”, Ann. Math. (2), 84 (1966), 157–174 | DOI | MR | Zbl

[28] D. Schneider, M. Weber, “Une remarque sur un théorème de Bourgain”, Séminaire de Probabilités, XXVII, Lecture Notes in Math., 1557, Springer, Berlin, 1993, 202–206 | DOI | MR | Zbl

[29] E. M. Stein, “On limits of sequences of operators”, Ann. of Math., 74 (1961), 140–170 | DOI | MR | Zbl

[30] M. Talagrand, Upper and Lower Bounds for Stochastic Processes, Erg. der Math. und ihrer Grenzgeb., 60, Springer, Berlin–Heidelberg, 2014 | MR | Zbl

[31] M. Talagrand, “Applying a theorem of Fernique”, Ann. Inst. H. Poincaré, 32 (1996), 779–799 | MR | Zbl

[32] M. Weber, Dynamical Systems and Processes, IRMA Lectures in Math. and Theor. Physics, 14, Eur. Math. Soc. Pub. House, 2009 | MR | Zbl

[33] M. Weber, Entropie métrique et convergence presque partout, Travaux en Cours, 58, Hermann, Paris, 1998 | MR

[34] M. Weber, “Entropy numbers in $L^p$-spaces for averages of rotations”, J. Math. Kyoto Univ., 37:4 (1997), 689–700 | MR | Zbl

[35] M. Weber, “The Stein randomization procedure”, Rendiconti Math., Ser. VII, 16 (1996), 569–605 | MR

[36] M. Weber, “Coupling of the GB set property for ergodic averages”, J. Theoret. Probab., 9:1 (1996), 105–112 | DOI | MR | Zbl

[37] M. Weber, “GB and GC sets in ergodic theory”, Probability in Banach spaces 9 (Sandjberg, 1993), Progr. Probab., 35, Birkhäuser Boston, Boston, MA, 1994, 129–151 | MR | Zbl

[38] M. Weber, “Opérateurs réguliers sur les espaces $L^p$”, Séminaire de probabilités, XXVII, Lectures Notes in Math., 1557, Springer, 1993, 207–215 | DOI | MR | Zbl