Orbits of certain endomorphisms of nilmanifolds and Hausdorff dimension
  We prove that for certain endomorphisms of a nilmanifold N the set S of those points such that the closure of its (forward) orbit contains no periodic points is large in the sense that for any non-empty open set U, the set U\cap S is of full Hausdorff dimension. When the manifold N is a torus, this result is due to S.G.Dani. 
C.S.Aravida, and Parameswaran Sankaran