1,783 research outputs found
Symmetry of entropy in higher rank diagonalizable actions and measure classification
An important consequence of the theory of entropy of Z-actions is that the
events measurable with respect to the far future coincide (modulo null sets)
with those measurable with respect to the distant past, and that measuring the
entropy using the past will give the same value as measuring it using the
future. In this paper we show that for measures invariant under multiparameter
algebraic actions if the entropy attached to coarse Lyapunov foliations fail to
display a stronger symmetry property of a similar type this forces the measure
to be invariant under non-trivial unipotent groups. Some consequences of this
phenomenon are noted
Graph Eigenfunctions and Quantum Unique Ergodicity
We apply the techniques of our previous paper to study joint eigenfunctions
of the Laplacian and one Hecke operator on compact congruence surfaces, and
joint eigenfunctions of the two partial Laplacians on compact quotients of
. In both cases, we show that quantum limit
measures of such sequences of eigenfunctions carry positive entropy on almost
every ergodic component. Together with prior work of the second named author,
this implies Quantum Unique Ergodicity for such functions.Comment: 8 page
The Higher Connectivity of Intersections of Real Quadrics
A linear system of real quadratic forms defines a real projective variety.
The real non-singular locus of this variety (more precisely of the underlying
scheme) has a highly connected double cover as long as each non-zero form in
the system has sufficiently high Witt index.Comment: 13 page
- …