Abstract
We propose to study multiple ergodic averages from multifractal analysis point of view. In some special cases in the symbolic dynamics, the Hausdorff dimensions of the level sets for the limit of these multiple ergodic averages are determined by using Riesz products.
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References
Billingsley P.: Ergodic theory and information. Wiley, New York (1965)
Bourgain J.: Double recurrence and almost sure convergence. J. Reine. Angew. Math. 404, 140–161 (1990)
Falconer KJ.: Fractal Geometry : Mathematical Foundations and Applications. 2nd edn. Wiley, New York (2003)
Fan AH.: Sur les dimensions de mesures. Studia Math. 111, 1–17 (1994)
Fan AH.: Individual behaviors of oriented walks. Stoc. Proc. App. 90, 263–275 (2000)
Fan AH., Feng DJ.: On the distribution of long-term time averages on symbolic space. J. Stat. Phys. 99(3–4), 813–856 (2000)
Fan AH., Feng DJ., Wu J.: Recurrence, dimension and entropy. J. London Math. Soc. (2) 64(1), 229–244 (2001)
Fan AH., Liao LM., Peyrière J.: Generic points in systems of specification and Banach valued Birkhoff ergodic average. Discrete Contin. Dyn. Syst. 21, 1103–1128 (2008)
Fan AH., Schmeling J., Wu M.: Multifractal analysis of multiple ergodic averages. C. R. Acad. Sci. Paris, Ser. I 349, 961–964 (2011)
Furstenberg H.: Ergodic behavior of diagonal measures and a theorem of Szemerédi on arithmetic progressions. J. d’Analyse Math. 31, 204–256 (1977)
Hewitt E., Zuckerman HS.: Singular measures with absolutely continuous convolution squares. Proc. Camb. Phil. Soc. 62, 399–420 (1966)
Host B., Kra K.: Nonconventional ergodic averages and nilmanifolds. Ann. Math. 161, 397–488 (2005)
Kenyon, R., Peres, Y., Solomyak, B.: Hausdorff dimension for fractals invariant under the multiplicative integers. Ergod. Theory Dynam. Syst. (2011, to appear)
Ma J.H., Wen Z.Y.: Besicovitch subsets of self-similar sets. Ann. Inst. Fourier (Grenoble) 52, 1061–1074 (2002)
Olivier E.: Multifractal analysis in symbolic dynamics and distribution of pointwise dimension for g-measures. Nonlinearity 12, 1571–1585 (1999)
Pesin Y.: Dimension theory in dynamical systems: contemporary views and applications. University of Chicago Press, Chicago (1997)
Schmeling J.: On the completeness of multifractal spectra. Ergod. Theory Dynam. Syst. 19(6), 1595–1616 (1999)
Zygmund A.: Trigonometric series, vols. I, II, 2nd edn. Cambridge University Press, New York (1959)
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Communicated by K. Schmidt.
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Fan, AH., Liao, L. & Ma, JH. Level sets of multiple ergodic averages. Monatsh Math 168, 17–26 (2012). https://doi.org/10.1007/s00605-011-0358-5
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DOI: https://doi.org/10.1007/s00605-011-0358-5