ar X iv : 0 90 6 . 35 67 v 1 [ m at h . D S ] 1 9 Ju n 20 09 Cascades and ε - invisibility ∗

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 ε ≪ 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 Lipshitz 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 εinvisible set and a uniform moderate (< 100) Lipshitz constant independent on ε. These dynamical systems from [IN] are also | ln ε|-distant from structurally unstable dynamical systems (in the class S of skew products). Recall that a property of dynamical system is locally typical if every close system possesses it as well. The invisibility property is thus C-robust. 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. 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-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 ε with ε = 2−n k where 3k is the Hausdorff dimension of the phase space. Work partially supported by the grants NSF 0700973, RFBR 07-01-00017-a and carried out in Cornell University and Independent University of Moscow Cornell University, US; Moscow State and Independent Universities, Steklov Math. Institute, Moscow Moscow State University, Independent University of Moscow, Laboratoire J.-V. Poncelet (UMI 2615)