Size of exceptional sets in weakly mixing systems

TL;DR

This paper investigates the size and structure of exceptional sets in weakly mixing systems, providing explicit constructions, sharp bounds, and applications to various dynamical models.

Contribution

It constructs a universal exceptional set with logarithmic bounds for weakly mixing systems and establishes sharpness of these bounds, advancing understanding of obstructions in ergodic theory.

Findings

Constructed a universal exceptional set with size bounds (n) for all systems in the class.

Proved the bounds are essentially sharp for certain tight maps without spacers.

Applied the principles to models like interval exchange transformations and substitution systems.

Abstract

We study exceptional sets for the Chacon transformation and, more generally, for a class of cutting-and-stacking transformations called restrictive tight maps. For these systems we explicitly construct a universal exceptional set \(J\subseteq\mathbb{N}\), valid uniformly for all measurable pairs \(A,B\in\mathscr{B}\), such that for every increasing function \(h:\mathbb{N}\to\mathbb{R}_{>0}\) diverging to infinity, \(|J\cap[0,n]|\le(\log n)^{h(n)}\) for all sufficiently large \(n\). The Chacon transformation considered in this paper belongs to this class, giving a logarithmic-scale universal exceptional set for Chacon. We also prove that this logarithmic scale is essentially sharp at the level of pairwise obstructions: for every tight map with no spacers above the last subcolumn, i.e. \(s_{m-1}=0\), and every \(t>0\), there exist measurable sets \(A,B\) such that every exceptional set \(J\) for \((A,B)\) satisfies \(|J\cap[0,n]|\ge(\log n)^t\) for all sufficiently large \(n\). The construction is based on recursive formulas for return-time distributions arising from the cutting-and-stacking structure. As a complementary quantitative principle, we show that if the corresponding \(p\)-th Ces\`aro weak-mixing averages satisfy a rate \(o(b_N)\), then \(J_{A,B}\) may be chosen so that \(|J_{A,B}\cap[0,N]|=o(Nb_N)\). We apply this rate-to-exceptional-set principle to several weakly mixing models, including interval exchange transformations, translation flows, and substitution dynamical systems, under the regularity assumptions of the available quantitative estimates. We also construct a separate weakly mixing one-spacer rank-one example in which exceptional-set obstructions have polynomial lower growth.