The group of invariants of an inner function with finite spectrum

TL;DR

This paper characterizes the group of continuous invariants of an inner function with finitely many singularities on the unit circle, identifying all continuous functions that leave the inner function unchanged under composition.

Contribution

It provides a complete description of the invariant group for inner functions with finitely many singularities, a problem not fully addressed before.

Findings

Identified the structure of the invariant group for such inner functions.

Characterized the continuous mappings that preserve the inner function.

Established conditions under which these invariants form a group.

Abstract

This paper determines the group of continuous invariants corresponding to an inner function $ \Theta $ with finitely many singularities on the unit circle $T$; that is, the continuous mappings $g: T \to T$ such that $\Theta \circ g = \Theta $ on $\T$. These mappings form a group under composition.