Iterative Roots of Multifunctions

TL;DR

This paper provides conditions under which multifunctions and certain complex polynomials lack iterative roots, focusing on graph structures with many paths to a point, with implications for single-valued maps.

Contribution

It introduces verifiable criteria for the nonexistence of iterative roots in multifunctions, extending to complex polynomials via their pullbacks.

Findings

Certain multifunctions with a point having many paths lack iterative roots.

Applied results to show some complex polynomials have no iterative roots of specified orders.

Abstract

Some easily verifiable sufficient conditions for the nonexistence of iterative roots for multifunctions on arbitrary nonempty sets are presented. Typically if the graph of the multifunction has a distinguished point with a relatively large number of paths leading to it then such a multifunction does not admit any iterative root. These results can be applied to single-valued maps by considering their pullbacks as multifunctions. This has been illustrated by showing the nonexistence of iterative roots of some specified orders for certain complex polynomials.