Only rational homology spheres admit $\Omega(f)$ to be union of DE attractors

TL;DR

This paper proves that only rational homology spheres can have a diffeomorphism with a non-wandering set composed of finitely many attractors derived from expanding maps, with all attractors being of dimension n-2.

Contribution

It establishes a topological restriction on manifolds admitting such dynamical systems, linking the structure of attractors to the manifold's homology.

Findings

Manifolds with such diffeomorphisms are rational homology spheres.

All attractors in these systems are of topological dimension n-2.

Expanding maps act on (co)homologies, influencing the manifold's topology.

Abstract

If there exists a diffeomorphism $f$ on a closed, orientable $n$-manifold $M$ such that the non-wandering set $\Omega(f)$ consists of finitely many orientable $(\pm)$ attractors derived from expanding maps, then $M$ must be a rational homology sphere; moreover all those attractors are of topological dimension $n-2$. Expanding maps are expanding on (co)homologies.