A Generalized Cross Ratio

TL;DR

This paper introduces a generalized cross ratio extending the classical concept from complex analysis, exploring its properties and conditions for well-definition, and examining related transitivity properties of certain maps.

Contribution

It defines a new generalized cross ratio, analyzes its properties, and identifies conditions for its well-definition, extending classical invariance concepts to broader map classes.

Findings

Defined a generalized cross ratio with specific properties.

Identified conditions for the cross ratio to be well-defined.

Established transitivity properties for a class of maps similar to the classical case.

Abstract

In one complex variable, the cross ratio is a well-known quantity associated with four given points in the complex plane that remains invariant under linear fractional maps. In particular, if one knows where three points in the complex plane are mapped under a linear fractional map, one can use this invariance to explicitly determine the map and to show that linear fractional maps are $3$-transitive. In this paper, we define a generalized cross ratio and determine some of its basic properties. In particular, we determine which hypotheses must be made to guarantee that our generalized cross ratio is well defined. We thus obtain a class of maps that obey similar transitivity properties as in one complex dimension, under some more restrictive conditions.