Quantum ergodicity for graphs related to interval maps

We prove quantum ergodicity for a family of graphs that are obtained from ergodic one-dimensional maps of an interval using a procedure introduced by Pakonski et al (J. Phys. A, v. 34, 9303-9317 (2001)). As observables we take the L^2 functions on the interval. The proof is based on the periodic orbit expansion of a majorant of the quantum variance. Specifically, given a one-dimensional, Lebesgue-measure-preserving map of an interval, we consider an increasingly refined sequence of partitions of the interval. To this sequence we associate a sequence of graphs, whose directed edges correspond to elements of the partitions and on which the classical dynamics approximates the Perron-Frobenius operator corresponding to the map. We show that, except possibly for subsequences of density 0, the eigenstates of the quantum graphs equidistribute in the limit of large graphs. For a smaller class of observables we also show that the Egorov property, a correspondence between classical and quantum evolution in the semiclassical limit, holds for the quantum graphs in question.Comment: 20 pages, 1 figur

Co-contractions of Graphs and Right-angled Artin Groups

We define an operation on finite graphs, called co-contraction. By showing that co-contraction of a graph induces an injective map between right-angled Artin groups, we exhibit a family of graphs, without any induced cycle of length at least 5, such that the right-angled Artin groups on those graphs contain hyperbolic surface groups. This gives the negative answer to a question raised by Gordon, Long and Reid.Comment: 18 pages, 8 figure

Configuration space integrals for embedding spaces and the Haefliger invariant

Let K be the space of long j-knots in R^n. In this paper we introduce a graph complex D and a linear map I from D to the de Rham complex of K via configuration space integral, and prove that (1) when both n>j>=3 are odd, the map I is a cochain map if restricted to graphs with at most one loop component, (2) when n-j>=2 is even, the map I is a cochain map if restricted to tree graphs, and (3) when n-j >=3 is odd, the map I added a correction term produces a (2n-3j-3)-cocycle of K which gives a new formulation of the Haefliger invariant when n=6k, j=4k-1 for some k.Comment: 41 pages, many figures (v2: Theorem 1.3 and its proof have been improved. many minor corrections. v3: Theorem 1.3 and its proof have been revised, since Lemma 5.26 in v2 was wrong. v4: The proof of Theorem 1.3 has been fully revised. To appear in J. Knot Theory Ramifications