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Asymptotics of the Scenery Flow and Properties of the Fourier Dimension

Ekström, Fredrik LU (2016)
Abstract
This thesis is about the scenery flow and the Fourier dimension.

The scenery flow is a semiflow on the space of Borel probability measures on the interval [-1, 1] that have the origin in their support. The flow acts on such a measure by enlarging it, restricting it back to [-1, 1] and renormalising it to a probability measure. A measure induces a family of probability distributions on the space of measures, given by the uniform distributions on the initial segments of the orbit under the scenery flow. Paper A is about the relation between the distributions induced by a measure and by its image under a diffeomorphism. Sufficient conditions, and a necessary condition, are given for the two frequency distributions to be asymptotic,... (More)
This thesis is about the scenery flow and the Fourier dimension.

The scenery flow is a semiflow on the space of Borel probability measures on the interval [-1, 1] that have the origin in their support. The flow acts on such a measure by enlarging it, restricting it back to [-1, 1] and renormalising it to a probability measure. A measure induces a family of probability distributions on the space of measures, given by the uniform distributions on the initial segments of the orbit under the scenery flow. Paper A is about the relation between the distributions induced by a measure and by its image under a diffeomorphism. Sufficient conditions, and a necessary condition, are given for the two frequency distributions to be asymptotic, and some examples are provided.

The Fourier dimension of a Borel probability measure on Euclidean space is defined via the asymptotic decay rate of the Fourier transform of the measure, in such a way that higher Fourier dimension means faster decay. The Fourier dimension of a Borel set is the supremum of the Fourier dimensions of all probability measures that are concentrated on the set. It is known that the Fourier dimension of a set is always less than or equal to the Hausdorff dimension of the set, and both equality and strict inequality are possible.

In Paper B it is shown that the Fourier dimension is not in general stable under countable unions of sets or finite convex combinations of measures, and some situations in which the Fourier dimension is stable are exhibited. A modification to the definition of the Fourier dimension is proposed, ensuring countable stability, and it is shown that the class of measures that have modified Fourier dimension at least s is characterised by their common null sets.

Paper C gives an example showing that the Fourier dimension is not stable under finite unions of sets.

Paper D shows, by constructing a random diffeomorphism, that every Borel set in the real line is diffeomorphic to a set whose Fourier and Hausdorff dimensions are equal. Moreover, the Fourier dimension is not invariant under
diffeomorphisms of class Cm for any finite m. (Less)
Please use this url to cite or link to this publication:
author
supervisor
opponent
  • Professor Pertti Mattila, University of Helsinki, Finland
organization
publishing date
type
Thesis
publication status
published
subject
pages
129 pages
defense location
Lecture hall MA:6, Annexet, Sölvegatan 20, Lund University, Faculty of Engineering
defense date
2016-11-04 13:15
ISBN
978-91-7623-970-4
978-91-7623-969-8
language
English
LU publication?
yes
id
722ecc30-1fb8-41f8-be5a-b18bf6dfa366
date added to LUP
2016-10-10 11:28:01
date last changed
2016-10-21 10:50:50
@phdthesis{722ecc30-1fb8-41f8-be5a-b18bf6dfa366,
  abstract     = {This thesis is about the scenery flow and the Fourier dimension.<br/><br/>The scenery flow is a semiflow on the space of Borel probability measures on the interval [-1, 1] that have the origin in their support. The flow acts on such a measure by enlarging it, restricting it back to [-1, 1] and renormalising it to a probability measure. A measure induces a family of probability distributions on the space of measures, given by the uniform distributions on the initial segments of the orbit under the scenery flow. Paper A is about the relation between the distributions induced by a measure and by its image under a diffeomorphism. Sufficient conditions, and a necessary condition, are given for the two frequency distributions to be asymptotic, and some examples are provided.<br/><br/>The Fourier dimension of a Borel probability measure on Euclidean space is defined via the asymptotic decay rate of the Fourier transform of the measure, in such a way that higher Fourier dimension means faster decay. The Fourier dimension of a Borel set is the supremum of the Fourier dimensions of all probability measures that are concentrated on the set. It is known that the Fourier dimension of a set is always less than or equal to the Hausdorff dimension of the set, and both equality and strict inequality are possible.<br/><br/>In Paper B it is shown that the Fourier dimension is not in general stable under countable unions of sets or finite convex combinations of measures, and some situations in which the Fourier dimension is stable are exhibited. A modification to the definition of the Fourier dimension is proposed, ensuring countable stability, and it is shown that the class of measures that have modified Fourier dimension at least s is characterised by their common null sets.<br/><br/>Paper C gives an example showing that the Fourier dimension is not stable under finite unions of sets.<br/><br/>Paper D shows, by constructing a random diffeomorphism, that every Borel set in the real line is diffeomorphic to a set whose Fourier and Hausdorff dimensions are equal. Moreover, the Fourier dimension is not invariant under<br/>diffeomorphisms of class C<sup>m</sup> for any finite m.},
  author       = {Ekström, Fredrik},
  isbn         = {978-91-7623-970-4},
  language     = {eng},
  month        = {10},
  pages        = {129},
  school       = {Lund University},
  title        = {Asymptotics of the Scenery Flow and Properties of the Fourier Dimension},
  year         = {2016},
}