Cascades of e-invisibility

Abstract

We consider statistical attractors of locally typical dynamical systems and their "e-invisible" subsets: parts of the attractors whose neighborhoods are visited by orbits with an average frequency of less than e << 1. For extraordinarily small values of e (say, smaller than 2^(-10^6)), an observer virtually never sees these parts when following a generic orbit. A trivial reason for e-invisibility in a generic dynamical system may be either a high Lipshitz constant (~1/e) of the mapping (i.e. it badly distorts the metric) or its proximity (~e) to the structurally unstable dynamical systems. However Ilyashenko and Negut [IN] provided a locally typical example of dynamical systems with an e-invisible set and a uniform moderate (<100) Lipshitz constant independent on e. These dynamical systems from [IN] are also |log e|^{-1}-distant from structurally unstable dynamical systems (in the class S of skew products). We further develop the example of [IN] to provide a better rate of invisibility while staying at the same distance away from the structurally unstable dynamical systems. We give an explicit example of C^1-balls in the space of "step" skew products over the Bernoulli shift such that for each dynamical system from this ball a large portion of the statistical attractor is invisible. Systems that are c/n-distant from structurally unstable ones (in the class S) have rate of invisibility e = 2^(-n^k) where 3k is the Hausdorff dimension of the phase space.