A nonlinear version of the $\alpha$-Kakutani equidistribution problem

TL;DR

This paper extends classical results on interval-splitting equidistribution to a nonlinear setting using contractions, establishing Lebesgue equidistribution under specific regularity conditions.

Contribution

It introduces a nonlinear framework for the $ ext{α}$-Kakutani problem and proves Lebesgue equidistribution with new regularity assumptions.

Findings

Proves Lebesgue equidistribution for nonlinear interval-splitting procedures.

Extends classical linear results to nonlinear contractions.

Establishes conditions under which equidistribution holds.

Abstract

In this work, we extend results of Kakutani; Adler and Flatto; Smilansky; Pollicott and Sewell on the equidistribution of endpoints generated by interval-splitting procedures. We study a nonlinear version of the problem generated by a finite or countable family of $\mathcal{C}^{1+ \varepsilon}$ contractions and prove Lebesgue equidistribution under suitable nonlattice and thermodynamic regularity assumptions.