11,526 research outputs found
Fourier transforms of Gibbs measures for the Gauss map
We investigate under which conditions a given invariant measure for the
dynamical system defined by the Gauss map is a Rajchman
measure with polynomially decaying Fourier transform We show that this
property holds for any Gibbs measure of Hausdorff dimension greater than
with a natural large deviation assumption on the Gibbs potential. In
particular, we obtain the result for the Hausdorff measure and all Gibbs
measures of dimension greater than on badly approximable numbers, which
extends the constructions of Kaufman and Queff\'elec-Ramar\'e. Our main result
implies that the Fourier-Stieltjes coefficients of the Minkowski's question
mark function decay to polynomially answering a question of Salem from
1943. As an application of the Davenport-Erd\H{o}s-LeVeque criterion we obtain
an equidistribution theorem for Gibbs measures, which extends in part a recent
result by Hochman-Shmerkin. Our proofs are based on exploiting the nonlinear
and number theoretic nature of the Gauss map and large deviation theory for
Hausdorff dimension and Lyapunov exponents.Comment: v3: 29 pages; peer-reviewed version, fixes typos and added more
elaborations, and included comments on Salem's problem. To appear in Math.
An
Properties of measures supported on fat Sierpinski carpets
In this paper we study certain conformal iterated function schemes in two dimensions that are natural generalizations of the Sierpinski carpet construction. In particular, we consider scaling factors for which the open set condition fails. For such āfat Sierpinski carpetsā we study the range of parameters for which the dimension of the set is exactly known, or for which the set has positive measure
What is tested when experiments test that quantum dynamics is linear
Experiments that look for nonlinear quantum dynamics test the fundamental
premise of physics that one of two separate systems can influence the physical
behavior of the other only if there is a force between them, an interaction
that involves momentum and energy. The premise is tested because it is the
assumption of a proof that quantum dynamics must be linear. Here variations of
a familiar example are used to show how results of nonlinear dynamics in one
system can depend on correlations with the other. Effects of one system on the
other, influence without interaction between separate systems, not previously
considered possible, would be expected with nonlinear quantum dynamics. Whether
it is possible or not is subject to experimental tests together with the
linearity of quantum dynamics. Concluding comments and questions consider
directions our thinking might take in response to this surprising unprecedented
situation.Comment: 14 pages, Title changed, sentences adde
- ā¦