Historic set carries full hausdorff dimension
author
Abstract:
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.
similar resources
Invariant Sets with Zero Measure and Full Hausdorff Dimension
For a subshift of finite type and a fixed Hölder continuous function, the zero measure invariant set of points where the Birkhoff averages do not exist is either empty or carries full Hausdorff dimension. Similar statements hold for conformal repellers and two-dimensional horseshoes, and the set of points where the pointwise dimensions, local entropies, Lyapunov exponents, and Birkhoff averages...
full textA Critical Set with Nonnull Image Has Large Hausdorff Dimension
The question of how complicated a critical set must be to have a nonnull image is answered by relating its Hausdorff dimension to the (Holder) differentiability of the map. This leads to a new extension of the Morse-Sard Theorem. The main tool is an extended version of Morse's Lemma.
full textThe Newhouse Set Has a Positive Hausdorff Dimension
The Newhouse phenomenon of infinitely many coexisting periodic attractors is studied in its simplest form. One shows that the corresponding parameter set (the Newhouse set) JN has a strictly positive Hausdorff dimension. This result is stronger than that of Tedeschini-Lalli and Yorke [Commun. Math. Phys. 106, 635 (1986)] concerning the Lebesgue measure of the Newhouse set; and is complementary ...
full textSets of “non-typical” Points Have Full Topological Entropy and Full Hausdorff Dimension
For subshifts of finite type, conformal repellers, and conformal horseshoes, we prove that the set of points where the pointwise dimensions, local entropies, Lyapunov exponents, and Birkhoff averages do not exist simultaneously, carries full topological entropy and full Hausdorff dimension. This follows from a much stronger statement formulated for a class of symbolic dynamical systems which in...
full textEffective Hausdorff Dimension
We continue the study of effective Hausdorff dimension as it was initiated by LUTZ. Whereas he uses a generalization of martingales on the Cantor space to introduce this notion we give a characterization in terms of effective s-dimensional Hausdorff measures, similar to the effectivization of Lebesgue measure by MARTIN-LÖF. It turns out that effective Hausdorff dimension allows to classify sequ...
full textEffective Hausdorff dimension
Lutz (2000) has recently proved a new characterization of Hausdorff dimension in terms of gales, which are betting strategies that generalize martingales. We present here this characterization and give three instances of how it can be used to define effective versions of Hausdorff dimension in the contexts of constructible, finite-state, and resource-bounded computation.
full textMy Resources
Journal title
volume 43 issue 7
pages 2339- 2347
publication date 2017-12-30
By following a journal you will be notified via email when a new issue of this journal is published.
Hosted on Doprax cloud platform doprax.com
copyright © 2015-2023