We study transitive step skew-product maps modeled over a complete shift of k, , symbols whose fiber maps are defined on the circle and have intermingled contracting and expanding regions. These dynamics are genuinely nonhyperbolic and exhibit simultaneously ergodic measures with positive, negative, and zero exponents.
We introduce a set of axioms for the fiber maps and study the dynamics of the resulting skew-product. These axioms turn out to capture the key mechanisms of the dynamics of nonhyperbolic robustly transitive maps with compact central leaves.
Focusing on the nonhyperbolic ergodic measures (with zero fiber exponent) of these systems, we prove that such measures are approximated in the topology and in entropy by hyperbolic ones. We also prove that they are in the intersection of the convex hulls of the measures with positive fiber exponent and with negative fiber exponent. Our methods also allow us to perturb hyperbolic measures. We can perturb a measure with negative exponent directly to a measure with positive exponent (and vice-versa), however we lose some amount of entropy in this process. The loss of entropy is determined by the difference between the Lyapunov exponents of the measures.
Keywords: Entropy, Ergodic measures, Lyapunov exponents, Skew-product, Transitivity
@article{AIHPC_2017__34_6_1561_0,
author = {D{\'\i}az, L.J. and Gelfert, K. and Rams, M.},
title = {Nonhyperbolic step skew-products: {Ergodic} approximation},
journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
pages = {1561--1598},
year = {2017},
publisher = {Elsevier},
volume = {34},
number = {6},
doi = {10.1016/j.anihpc.2016.10.009},
zbl = {1475.37044},
mrnumber = {3712011},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.1016/j.anihpc.2016.10.009/}
}
TY - JOUR AU - Díaz, L.J. AU - Gelfert, K. AU - Rams, M. TI - Nonhyperbolic step skew-products: Ergodic approximation JO - Annales de l'I.H.P. Analyse non linéaire PY - 2017 SP - 1561 EP - 1598 VL - 34 IS - 6 PB - Elsevier UR - http://geodesic.mathdoc.fr/articles/10.1016/j.anihpc.2016.10.009/ DO - 10.1016/j.anihpc.2016.10.009 LA - en ID - AIHPC_2017__34_6_1561_0 ER -
%0 Journal Article %A Díaz, L.J. %A Gelfert, K. %A Rams, M. %T Nonhyperbolic step skew-products: Ergodic approximation %J Annales de l'I.H.P. Analyse non linéaire %D 2017 %P 1561-1598 %V 34 %N 6 %I Elsevier %U http://geodesic.mathdoc.fr/articles/10.1016/j.anihpc.2016.10.009/ %R 10.1016/j.anihpc.2016.10.009 %G en %F AIHPC_2017__34_6_1561_0
Díaz, L.J.; Gelfert, K.; Rams, M. Nonhyperbolic step skew-products: Ergodic approximation. Annales de l'I.H.P. Analyse non linéaire, Tome 34 (2017) no. 6, pp. 1561-1598. doi: 10.1016/j.anihpc.2016.10.009
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