Skew-product maps with base having closed set of periodic points

In [Proc. ECIT-89, World Scientific, (1991), 177–183], A. N. Sharkovski˘ı and S.F. Kolyada stated the problem of characterization skew-product maps having zero topological entropy. It is known that, even under some additional assumptions, this aim has not been reached. In [J. Math. Anal. Appl., 287, (2003), 516–521], J. L. G. Guirao and J. Chudziak partially solved the problem in the class of skew-product maps with base map having closed set of periodic points. The present paper has two aims for this class of maps, on one hand to improve that solution showing the equivalence between the property “to have zero topological entropy” and the fact “not to be Li-Yorke chaotic in the union of the ω-limit sets of recurrent points”. On other hand, we show that the properties “to have closed set of periodic points” and “all nonwandering points are periodic” are not mutually equivalent properties, for doing this we disprove a result from Efremova of 1990

Nonhyperbolic step skew-products: Ergodic approximation

We study transitive step skew-product maps modeled over a complete shift of $k$, $k\ge2$, 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 weak$\ast$ 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.Comment: 43 pages, 5 figure