A toral automorphism is a linear map on the N-dimensional torus defined by an integer matrix. Ergodicity means almost every orbit is equidistributed. Stable ergodicity means that every C¹-small perturbation is also ergodic — the ergodicity is robust.
The paper on stable ergodicity of toral automorphisms (arXiv: 2603.23778) proves that all ergodic automorphisms with two-dimensional center are stably ergodic. This includes all ergodic automorphisms in dimensions N ≤ 5 and N = 7.
The center direction is the hard part. An automorphism decomposes the tangent space into expanding, contracting, and center directions. The expanding and contracting parts are hyperbolic and behave well. The center — where the eigenvalues lie on the unit circle — is where ergodicity can fail under perturbation. Previous results required algebraic conditions on the characteristic polynomial to control the center. This paper removes those conditions when the center is two-dimensional.
The proof's core is a minimality criterion: showing that certain foliations have all leaves dense, which forces equidistribution. The criterion works because in two center dimensions, the topology constrains the foliation enough.
The through-claim: two center dimensions is the threshold where topology controls dynamics. In one center dimension, stable ergodicity was known. In two, the algebraic conditions can be removed. The topological structure of two-dimensional center foliations — their leaves must be dense for geometric reasons — does the work that algebra did before.
2603.23778. Dynamical systems / stable ergodicity / toral automorphisms / partially hyperbolic / center foliation.