78 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
Invariant measures and the set of exceptions to Littlewood's conjecture
We classify the measures on SL (k,R)/SL (k,Z) which are invariant and ergodic
under the action of the group A of positive diagonal matrices with positive
entropy. We apply this to prove that the set of exceptions to Littlewood's
conjecture has Hausdorff dimension zero.Comment: 48 page
Entropy of convolutions on the circle
Given ergodic p-invariant measures {\mu_i} on the 1-torus T=R/Z, we give a
sharp condition on their entropies, guaranteeing that the entropy of the
convolution \muon converges to \log p. We also prove a variant of this result
for joinings of full entropy on \T^\N. In conjunction with a method of Host,
this yields the following. Denote \sig_q(x) = qx\pmod{1}. Then for every
p-invariant ergodic \mu with positive entropy,
\frac{1}{N}\sum_{n=0}^{N-1}\sig_{c_n}\mu converges weak^* to Lebesgue measure
as N \goesto \infty, under a certain mild combinatorial condition on {c_k}.
(For instance, the condition is satisfied if p=10 and c_k=2^k+6^k or
c_k=2^{2^k}.) This extends a result of Johnson and Rudolph, who considered the
sequence c_k = q^k when p and q are multiplicatively independent.
We also obtain the following corollary concerning Hausdorff dimension of sum
sets: For any sequence {S_i} of p-invariant closed subsets of T, if \sum
\dim_H(S_i) / |\log\dim_H(S_i)| = \infty, then \dim_H(S_1 + \cdots + S_n)
\goesto 1.Comment: 34 pages, published versio
Quantum Ergodicity and Averaging Operators on the Sphere
We prove quantum ergodicity for certain orthonormal bases of
, consisting of joint eigenfunctions of the Laplacian on
and the discrete averaging operator over a finite set of
rotations, generating a free group. If in addition the rotations are algebraic
we give a quantified version of this result. The methods used also give a new,
simplified proof of quantum ergodicity for large regular graphs.Comment: 27 page
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