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To indicate the statistical complexity of dynamical systems, we introduce the notions of higher order irregular set and higher order maximal Birkhoff average oscillation in this paper. We prove that, in the setting of topologically mixing Markov chain, the set consisting of those points having maximal $k$-order Birkhoff average oscillation for all positive integers $k$ is as large as the whole space from the topological point of view. As applications, we discuss the corresponding results on a repeller.
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참고문헌 (27)
참고문헌 신청
A. H. Fan / 2000 / On the distribution of long-term time averages on symbolic space / J. Stat. Phys 99 (3-4) : 813 ~ 856
A. H. Fan / 2001 / Recurrence, dimensions and entropies / J. London Math. Soc 64 (1) : 229 ~ 244
C. Bonatti / 2005 / Dynamics Beyond Uniform Hyperbolicity / Springer-Verlag
D. Ruelle / 1978 / Thermodynamic Formalism: The mathematical structures of classical equi-librium statistical mechanics / Addison Wesley
D. Thompson / 2010 / The irregular set for maps with the specification property has full topo-logical pressure / Dyn. Syst 25 (1) : 25 ~ 51
A. H. Fan / 2001 / Recurrence, dimensions and entropies / J. London Math. Soc 64 (1) : 229 ~ 244
C. Bonatti / 2005 / Dynamics Beyond Uniform Hyperbolicity / Springer-Verlag
D. Ruelle / 1978 / Thermodynamic Formalism: The mathematical structures of classical equi-librium statistical mechanics / Addison Wesley
D. Thompson / 2010 / The irregular set for maps with the specification property has full topo-logical pressure / Dyn. Syst 25 (1) : 25 ~ 51
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