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

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 ${\mathcal{C}}^{1+\epsilon }$ contractions and prove Lebesgue equidistribution under suitable nonlattice and thermodynamic regularity assumptions.