Abstract:
A strengthened version of the Birkhoff ergodic theorem for linear Anosov diffeomorphisms on the 2-torus
is presented. Namely, it is proved that the Hausdorff dimension of the set of points at which partial limits of the time average strongly (by a constant) differ from the space average is strictly less than the dimension of the space (i.e., than 2).
Keywords:
Hausdorff dimension, special ergodic theorem, Anosov diffeomorphism, Markov chain, large deviation theorem.
Citation:
P. S. Saltykov, “A Special Ergodic Theorem for Anosov Diffeomorphisms on the 2-Torus”, Funktsional. Anal. i Prilozhen., 45:1 (2011), 69–78; Funct. Anal. Appl., 45:1 (2011), 56–63
\Bibitem{Sal11}
\by P.~S.~Saltykov
\paper A Special Ergodic Theorem for Anosov Diffeomorphisms on the 2-Torus
\jour Funktsional. Anal. i Prilozhen.
\yr 2011
\vol 45
\issue 1
\pages 69--78
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\crossref{https://doi.org/10.4213/faa3028}
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\transl
\jour Funct. Anal. Appl.
\yr 2011
\vol 45
\issue 1
\pages 56--63
\crossref{https://doi.org/10.1007/s10688-011-0006-9}
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Linking options:
https://www.mathnet.ru/eng/faa3028
https://doi.org/10.4213/faa3028
https://www.mathnet.ru/eng/faa/v45/i1/p69
This publication is cited in the following 6 articles:
Ilyashenko Yu. Shilin I., “Attractors and Skew Products”, Modern Theory of Dynamical Systems: a Tribute to Dmitry Victorovich Anosov, Contemporary Mathematics, 692, ed. Katok A. Pesin Y. Hertz F., Amer Mathematical Soc, 2017, 155–175
N. A. Solodovnikov, “Boundary-preserving mappings of a manifold with intermingling basins of components of the attractor, one of which is open”, Trans. Moscow Math. Soc., 75 (2014), 69–76
Kleptsyn V. Ryzhov D. Minkov S., “Special ergodic theorems and dynamical large deviations”, Nonlinearity, 25:11 (2012), 3189–3196
Ilyashenko Yu. Negut A., “Hölder properties of perturbed skew products and Fubini regained”, Nonlinearity, 25:8 (2012), 2377–2399
V. A. Kleptsyn, P. S. Saltykov, “On C2-stable effects of intermingled basins of attractors in classes of boundary-preserving maps”, Trans. Moscow Math. Soc., 72 (2011), 193–217
Citation in format AMSBIB
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