We consider statistical attractors of locally typical dynamical systems and their “ε-invisible” subsets: parts of the attractors whose neighborhoods are visited by orbits with an average frequency of less than $${\varepsilon \ll 1}$$ . For extraordinarily small values of ε (say, smaller than $${{2^{-10}}^6}$$ ), an observer virtually never sees these parts when following a generic orbit.
A trivial reason for ε-invisibility in a generic dynamical system may be either a high Lipschitz constant (~ 1/ε) of the mapping (i.e. it badly distorts the metric) or its close (~ ε) proximity to the structurally unstable dynamical systems. However, [IN] provided a locally typical example of dynamical systems with an ε-invisible set and a uniform moderate (< 100) Lipschitz constant independent of ε. These dynamical systems from [IN] are also |log ε|-distant from the structurally unstable dynamical systems (in the class $${\mathcal{S}}$$ of skew products). Recall that a property of dynamical systems is locally typical if every close system possesses it as well. The invisibility property is thus C 1-robust.
We further develop the example of [IN] to provide a better rate of invisibility while keeping the same radius of the ball in the space of skew products. Our construction is based on series of cascading dynamical systems. Each system incorporates the previous one and further boosts the invisibility rate. 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. The systems have rate of invisibility $${\varepsilon = {2^{-n}}^k}$$ .