Arithmetic Quantum Unique Ergodicity for Symplectic Linear Maps of the Multidimensional Torus

The quantization of linear automorphisms of the torus, is an arithmetic model for a quantum system with underlying chaotic classical dynamics. This model was studied over the last three decades, in an attempt to gain better understanding of phenomena in quantum chaos. In this thesis, we study a multidimensional analogue of this model. This multidimensional model exhibits some new phenomena that did not occur in the original (two dimensional) model. The classical dynamics underlying the model, is a discrete time dynamics given by the action of a symplectic linear map A ∈ Sp(2d,Z) on a torus T = R/Z. This dynamical system is ergodic and mixing, and presents a good model for chaotic dynamics. The quantization of this system, introduced by Hannay and Berry, consists of a family of finite dimensional Hilbert spaces of states HN (of dimensions N), together with unitary operators UN(A) acting on HN referred to as the quantum propagator. The semiclassical limit in this model is achieved by taking N →∞. Any quantum state can be interpreted as a distribution on the torus, by considering the corresponding matrix element of a quantum observable. For a stationary state (i.e., an eigenstate of the quantum propagator), this distribution is invariant under the classical dynamics. The Quantum Ergodicity Theorem, states that in the semi-classical limit almost all of the quantum distributions (corresponding to stationary states), converge to Lebesgue measure on the torus. The system is said to be Quantum Uniquely Ergodic, if the only limiting measure obtained from stationary states is the Lebesgue measure. For the two dimensional model (d = 1), after taking into account certain arithmetic symmetries, Kurlberg and Rudnick showed that the only limiting measure is the volume measure (this notion is referred to as Arithmetic Quantum Unique Ergodicity). With out taking the arithmetic symmetries into account, this is no longer true. Indeed, Faure, Nonnenmacher and De Bièvre demonstrated the existence of scars, a subsequence of eigenfunctions for which the corresponding distributions concentrate around a periodic orbit. In this thesis, we study the multidimensional model (d > 1). We show that for a symplectic linear map that leaves no invariant isotropic rational subspaces, similar to the two dimensional model, the system is Arithmetically QUE. However, if there are invariant isotropic rational subspaces, then the induced system is no longer Arithmetically QUE. To show this, we demonstrate the existence of super-scars, limiting measures (that are stable under the arithmetic symmetries) and are localized on an invariant sub-manifold. This thesis includes several more results concerning the fluctuations of the matrix el-