Stable arcs connecting polar cascades on a torus

TL;DR

This paper solves a longstanding problem in dynamical systems by characterizing stable arcs connecting polar cascades on a torus, specifically for gradient-like diffeomorphisms with fixed non-wandering points.

Contribution

It provides a solution to the 33rd Palis-Pugh problem for a specific class of polar gradient-like diffeomorphisms on the torus, under certain fixed point conditions.

Findings

Established the existence of stable arcs connecting polar cascades.

Characterized the structure of polar gradient-like diffeomorphisms on a torus.

Extended understanding of non-wandering point configurations.

Abstract

In this paper, we obtain a solution to the 33rd Palis-Pugh problem for polar gradient-like diffeomorphisms on a two-dimensional torus, under the assumption that all non-wandering points are fixed and have a positive orientation type.