Orbits of Certain Endomorphisms of Nilmanifolds and Hausdorff Dimension

Abstract

Let R be an element of GL(n, ℝ) having integer entries and let ρ : \({\mathbb{T}^n} \to {\mathbb{T}^n}\) denote the induced map on the torus \({\mathbb{T}^n} = {\mathbb{R}^n}/{\mathbb{Z}^n}\). It is well known that ρ is ergodic with respect to the Haar measure on \({\mathbb{T}^n}\) if and only if none of the eigenvalues of R is a root of unity. Dani has shown that there exists a subset S of \({\mathbb{T}^n}\) such that for any x ∈ S and any semisimple surjective endomorphism ρ of \({\mathbb{T}^n}\) such that the corresponding linear endomorphism has no eigenvalue on the unit circle, the closure of the orbit {ρ k (x)|k ≥ 0} contains no periodic points and that the set S is ‘large’ in the sense that for any nonempty open set U of \({\mathbb{T}^n}\) the set U ∩ S has Hausdorff dimension n. In this paper, we shall prove an analogous result for certain endomorphisms of nilmanifolds and infranil manifolds.