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Nontypical Behaviour of Orbits and Birkhoff Averages for Expanding Maps

Färm, David LU (2011)
Abstract
This thesis consists of an introductory chapter followed by five papers.



In the first paper, expanding maps on the unit interval are considered. The set of points for which the forward orbit is bounded away from a given point is studied. It is shown that this set has full Hausdorff dimension and that it has large intersection properties.



In the second paper, expanding maps on the unit interval with bounded distortion are considered. The set of points for which certain Birkhoff averages accumulate at a given value is studied. It is shown that this set has large intersection properties. As an application, it is shown that the set of points for which these Birkhoff averages do not converge has full... (More)
This thesis consists of an introductory chapter followed by five papers.



In the first paper, expanding maps on the unit interval are considered. The set of points for which the forward orbit is bounded away from a given point is studied. It is shown that this set has full Hausdorff dimension and that it has large intersection properties.



In the second paper, expanding maps on the unit interval with bounded distortion are considered. The set of points for which certain Birkhoff averages accumulate at a given value is studied. It is shown that this set has large intersection properties. As an application, it is shown that the set of points for which these Birkhoff averages do not converge has full Hausdorff dimension, even for countably many different maps simultaneously.



In the third paper, non-integer base expansions of numbers on the unit interval, are studied. For a dense set of bases, these expansions are generated by expanding maps for which the results from the first two papers of this thesis apply. In this paper, approximation arguments are used to extend these results to all bases.



In the fourth paper, attractors of conformal iterated function systems are considered. A family of classes of sets with large intersection properties, introduced by K. Falconer, is extended from Euclidean spaces without holes to this new fractal setting. As an application, the results from the second paper of this thesis are generalized.



In the fifth paper, a family of hyperbolic maps, similar to the fat baker's transformation, is studied. Depending on the parameters, these maps either expand or shrink area. It is shown, using the SRB-measures of the maps, that in the expanding case the attractor has positive Lebesgue measure for typical values of the parameters, while in the contracting case it tends to have Hausdorff dimension according to a certain formula. A key part of the proofs is the transversality of certain power series with coefficients from non-integer base expansions of real numbers. (Less)
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author
supervisor
opponent
  • Professor Pollicott, Mark, Mathematics Institute, University of Warwick, Coventry, United Kingdom
organization
publishing date
type
Thesis
publication status
published
subject
keywords
expanding maps, beta-shifts, large intersection classes, Hausdorff dimension
defense location
Lecture hall MH:C, Centre for Mathematical Sciences, Sölvegatan 18, Lund University Faculty of Engineering
defense date
2011-05-20 13:15
ISSN
1404-0034
ISBN
978-91-7473-099-9
language
English
LU publication?
yes
id
49527a5c-cbea-4bb1-b034-b0be1b91fc57 (old id 1895401)
date added to LUP
2011-04-27 09:25:22
date last changed
2016-09-19 08:45:00
@phdthesis{49527a5c-cbea-4bb1-b034-b0be1b91fc57,
  abstract     = {This thesis consists of an introductory chapter followed by five papers. <br/><br>
<br/><br>
In the first paper, expanding maps on the unit interval are considered. The set of points for which the forward orbit is bounded away from a given point is studied. It is shown that this set has full Hausdorff dimension and that it has large intersection properties.<br/><br>
<br/><br>
In the second paper, expanding maps on the unit interval with bounded distortion are considered. The set of points for which certain Birkhoff averages accumulate at a given value is studied. It is shown that this set has large intersection properties. As an application, it is shown that the set of points for which these Birkhoff averages do not converge has full Hausdorff dimension, even for countably many different maps simultaneously.<br/><br>
<br/><br>
In the third paper, non-integer base expansions of numbers on the unit interval, are studied. For a dense set of bases, these expansions are generated by expanding maps for which the results from the first two papers of this thesis apply. In this paper, approximation arguments are used to extend these results to all bases.<br/><br>
<br/><br>
In the fourth paper, attractors of conformal iterated function systems are considered. A family of classes of sets with large intersection properties, introduced by K. Falconer, is extended from Euclidean spaces without holes to this new fractal setting. As an application, the results from the second paper of this thesis are generalized. <br/><br>
<br/><br>
In the fifth paper, a family of hyperbolic maps, similar to the fat baker's transformation, is studied. Depending on the parameters, these maps either expand or shrink area. It is shown, using the SRB-measures of the maps, that in the expanding case the attractor has positive Lebesgue measure for typical values of the parameters, while in the contracting case it tends to have Hausdorff dimension according to a certain formula. A key part of the proofs is the transversality of certain power series with coefficients from non-integer base expansions of real numbers.},
  author       = {Färm, David},
  isbn         = {978-91-7473-099-9},
  issn         = {1404-0034},
  keyword      = {expanding maps,beta-shifts,large intersection classes,Hausdorff dimension},
  language     = {eng},
  school       = {Lund University},
  title        = {Nontypical Behaviour of Orbits and Birkhoff Averages for Expanding Maps},
  year         = {2011},
}