Addendum to “Singular equivariant asymptotics and Weyl’s law”

Let M be a closed Riemannian manifold carrying an effective and isometric action of a compact connected Lie group G. We derive a refined remainder estimate in the stationary phase approximation of certain oscillatory integrals on T^{\ast}M\times G with singular critical sets that were examined in [7] in order to determine the asymptotic distribution of eigenvalues of an invariant elliptic operator on M. As an immediate consequence, we deduce from this an asymptotic multiplicity formula for families of irreducible representations in \mathrm{L}^{2}(M) . The improved remainder is used in [4] to prove an equivariant semiclassical Weyl law and a corresponding equivariant quantum ergodicity theorem.

Quantum Limits of Eisenstein Series and Scattering States

We identify the quantum limits of scattering states for the modular surface. This is obtained through the study of quantum measures of non-holomorphic Eisenstein series away from the critical line. We provide a range of stability for the quantum unique ergodicity theorem of Luo and Sarnak.

Exponential Growth Rates of Periodic Asymmetric Oscillators

In this paper we will study the dynamics of the periodic asymmetric oscillator xʺ + qdoes exist for each non-zero solution x(t) of the oscillator. The properties of these rates, or the Lyapunov exponents, will be given using the induced circle di®eomorphism of the oscillator. The proof is extensively based on the Denjoy theorem in topological dynamics and the unique ergodicity theorem in ergodic theory.

On the dynamics of the linear action of SL(n, Z)

Using Moore's ergodicity theorem, S.G. Dani and S. Raghavan proved that the linear action of SL(n, ℤ) on ℝn is topologically (n − l)-transitive; that is, topologically transitive on the Cartesian product of n − 1 copies of ℝn. In this paper, we give a more direct proof, using the prime number theorem. Further, using the congruence subgroup theorem, we generalise the result to arbitrary finite index subgroups of SL(n, ℤ).

On records and related processes for sequences with trends

We study the records and related variables for sequences with linear trends. We discuss the properties of the asymptotic rate function and relationships between the distribution of the long-term maxima in the sequence and that of a particular observation, including two characterization type results. We also consider certain Markov chains related to the process of records and prove limit theorems for them, including the ergodicity theorem in the regular case (convergence rates are given under additional assumptions), and derive the limiting distributions for the inter-record times and increments of records.

On records and related processes for sequences with trends

We study the records and related variables for sequences with linear trends. We discuss the properties of the asymptotic rate function and relationships between the distribution of the long-term maxima in the sequence and that of a particular observation, including two characterization type results. We also consider certain Markov chains related to the process of records and prove limit theorems for them, including the ergodicity theorem in the regular case (convergence rates are given under additional assumptions), and derive the limiting distributions for the inter-record times and increments of records.