On Higher Rank Commutative Actions by Toral Automorphisms.

摘要

We study rigidity properties of Zr -actions by toral automorphisms. Such actions form an important class of examples of higher rank commutative algebraic actions, and are interesting from an arithmetic point of view as under certain assumptions they are conjugate to the natural action of a subgroup of units from a number field K on some compact quotient of K ⊗QR . In 1983, Berend proved that a faithful Zr -action alpha on Td by automorphisms is topologically rigid, which means any orbit is either finite or dense, if and only if r ≥ 2 and alpha is hyperbolic and contains totally irreducible toral automorphisms. In this thesis we present three different extensions to Berend's result.;When alpha satisfies Berend's conditions and is a Cartan action, i.e. alpha is not contained in a faithful commutative action of strictly higher rank by toral automorphisms, we show a quantitative version of its rigidity by generalizing Bourgain, Lindenstrauss, Michel and Venkatesh's recent one-dimensional result. We also give an application of our quantitative estimate to the geometry of numbers in number fields.;Next, we show that when alpha has higher rank and total irreducibility but fails to be hyperbolic, there is still rigidity in the weaker sense that any point has a dense orbit unless it lies in the central foliation through some rational point. More generally, the result applies to the partial action by those toral automorphisms from a Zr -action that are roughly isometric along a given set of eigenspaces.;The last part of this thesis represents a joint work with my advisor Prof. Elon Lindenstrauss, in which we investigate two-fold self-joinings of a Cartan action by toral automorphisms. We consider the diagonal Zr -action alpha▵ = alpha x alpha on ( T )2d, where alpha satisfies Berend's conditions and is Cartan. When r ≥ 3, we establish a rigidity property that asserts any orbit closure is homogeneous and has dimension 0, d or 2d. However for r = 2, we construct a counterexample of non-homogeneous orbit closure.