Historic set carries full hausdorff dimension

Document Type : Research Paper


School of Mathematics and Statistics‎, ‎Anyang Normal University‎, ‎Henan‎, ‎455000‎, ‎China.


‎We prove that the historic set for ratio‎ ‎of Birkhoff average is either empty or full of Hausdorff dimension in a class of one dimensional‎ ‎non-uniformly hyperbolic dynamical systems.


Main Subjects

L. Barreira, J.-J. Li and V. Claudia, Irregular sets are residual, Tohoku Math. J. 66 (2014), no. 4, 471--489.
L. Barreira and J. Schmeling, Sets of non-typical points have full topological entropy and full Hausdorff dimension, Israel J. Math. 116 (2000), no. 1, 29--70.
K. Falconer, Techniques in Fractal Geometry, John Wiley & Sons, Chichester, 1997.
A.-H. Fan, D.-J. Feng and J. Wu, Recurrence, dimension and entropy, J. Lond. Math. Soc. 64 (2001), no. 1, 229--244.
D.-J. Feng, K.-S. Lau and J. Wu, Ergodic limits on the conformal repellers, Adv. Math. 169 (2002), no. 1, 58--91.
A. Johansson, T. Jordan, A. Oberg and M. Pollicott, Multifractal analysis of non-uniformly hyperbolic systems, Israel J. Math. 177 (2010), no. 1, 125--144.
G.-Z. Ma and X. Yao, Higher dimensional multifractal analysis of non-uniformly hyperbolic systems, J. Math. Anal. Appl. 421 (2015), no. 1, 669--684.
 L. Olsen, Multifractal analysis of divergence points of deformed measure theoretical Birkhoff averages, J. Math. Pures Appl. 82 (2003), no. 12, 1591--1649.
K.E. Petersen, Ergodic Theory. Vol. 2. Cambridge Univ. Press, 1989.
D. Ruelle, Historical behaviour in smooth dynamical systems, in: Global Analysis of Dynamical Systems, pp. 63--66, CRC Press, 2001.
F. Takens, Orbits with historic behaviour, or non-existence of averages, Nonlinearity 21 (2008), no. 3, 33--36.
D. Thompson, The irregular set for maps with the specification property has full topological pressure, Dyn. Syst. 25 (2010), no. 1, 25--51.
D. Thompson, Irregular sets, the β-transformation and the almost specification property, Trans. Amer. Math. Soc. 364 (2012), no. 10, 5395--5414.
X.-T. Tian, Different asymptotic behavior versus same dynamical complexity: Recurrence & (ir)regularity, Adv. Math. 288 (2016), 464--526.
P. Walters, An introduction to ergodic theory. Grad. Texts in Math. 79, Springer-Verlag, New York-Berlin, 1982.
M. Urbanski, Parabolic Cantor sets, Fund. Math. 151 (1996), no. 3, 241--277.