The inverse problem for canonically bounded rank-one transformations

TL;DR

This paper characterizes when canonically bounded rank-1 transformations are isomorphic to their inverse, providing a simple necessary and sufficient condition based on cutting and spacer parameters.

Contribution

It establishes a complete characterization of canonically bounded rank-1 transformations that are isomorphic to their inverse, linking it to a specific condition on parameters.

Findings

The condition is both necessary and sufficient for isomorphism to the inverse.

Provides a simple criterion based on cutting and spacer parameters.

Clarifies the structure of canonically bounded rank-1 transformations.

Abstract

Given the cutting and spacer parameters for a rank-1 transformation, there is a simple condition which is easily seen to be sufficient to guarantee that the transformation under consideration is isomorphic to its inverse. Here we show that if the cutting and spacer parameters are canonically bounded, that condition is also necessary, thus giving a simple characterization of the canonically bounded rank-1 transformations that are isomorphic to their inverse.