Cascades of e-invisibility
Yulij Ilyashenko, Denis Volk

TL;DR
This paper explores the concept of e-invisible subsets within statistical attractors of dynamical systems, providing explicit examples of systems with highly invisible regions that are still close to structurally stable systems.
Contribution
It extends previous work by constructing explicit C^1-skew product examples with improved invisibility rates while maintaining proximity to structurally unstable systems.
Findings
Large portions of attractors can be e-invisible with e = 2^(-n^k)
Explicit C^1-skew product examples demonstrate high invisibility rates
Invisibility rate relates to Hausdorff dimension of phase space
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…
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Taxonomy
TopicsOpinion Dynamics and Social Influence
