The problem with twp linear branches

TL;DR

This paper investigates the invariant measures of two-branch linear piecewise fractional maps, providing a partial solution to a problem where explicit forms are known only for maps with three or more branches.

Contribution

It offers a partial solution to understanding invariant measures for two-branch linear fractional maps, a case not explicitly solved before.

Findings

Explicit forms for invariant measures are known for maps with three or more branches.

No explicit form is currently known for two-branch linear fractional maps.

The paper proposes a partial solution to this open problem.

Abstract

Piecewise fractional linear maps wzth three or more branches have been studied in several papers. For many Moebius maps the shape of the density of their invariant measurs can be written down exactly. However, if just two branches are linear, no explicit form is known. In this paper a partial solution is offered.