Knaster's problem for almost $(Z_p)^k$-orbits

TL;DR

This paper extends Knaster's problem by proving that for certain almost orbits of p-tori on spheres, there exists a rotation making a continuous map constant on those orbits, advancing understanding of symmetry in topological maps.

Contribution

It introduces new cases of Knaster's problem involving almost orbits of p-tori, demonstrating the existence of rotations that make continuous maps constant on these orbits.

Findings

Existence of rotations making maps constant on almost p-torus orbits

Extension of Knaster's problem to new orbit configurations

Application to maps into real line and plane

Abstract

In this paper some new cases of Knaster's problem on continuous maps from spheres are established. In particular, we consider an almost orbit of a $p$-torus $X$ on the sphere, a continuous map $f$ from the sphere to the real line or real plane, and show that $X$ can be rotated so that $f$ becomes constant on $X$.