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Dynamics of Smooth Hyperbolic Systems with Singularities

Persson, Tomas LU orcid (2006)
Abstract
This thesis is about dynamics of piecewise hyperbolic maps on bounded sets in the plane and on the interval. It is based on the following papers:



T. Persson, A piecewise hyperbolic map with absolutely continuous invariant measure, Dynamical Systems: An International Journal, 21:3 (2006), 363--378



T. Persson, Absolutely continuous invariant measures for some piecewise hyperbolic affine maps, Preprints in Mathematical Sciences 2005:31, LUTFMA-5066-2005, ISSN 1403-9338



T. Persson, J. Schmeling, Dyadic Diophantine Approximation and Katok's Horseshoe Approximation,



Preprints in Mathematical Sciences 2006:3, LUTFMA-5066-2006, ISSN 1403-9338



The first... (More)
This thesis is about dynamics of piecewise hyperbolic maps on bounded sets in the plane and on the interval. It is based on the following papers:



T. Persson, A piecewise hyperbolic map with absolutely continuous invariant measure, Dynamical Systems: An International Journal, 21:3 (2006), 363--378



T. Persson, Absolutely continuous invariant measures for some piecewise hyperbolic affine maps, Preprints in Mathematical Sciences 2005:31, LUTFMA-5066-2005, ISSN 1403-9338



T. Persson, J. Schmeling, Dyadic Diophantine Approximation and Katok's Horseshoe Approximation,



Preprints in Mathematical Sciences 2006:3, LUTFMA-5066-2006, ISSN 1403-9338



The first two papers are about two different classes of piecewise affine hyperbolic maps on bounded subsets of the plane. For both classes the tangent space can be decomposed into one expanding and one contracting subspace and the directions of those subspaces are the same for any point in the manifold. It is shown that, in the sense of parameters, for almost all maps in the two classes there exists an invariant measure that is absolutely continuous with respect to Lebesgue measure, provided that the map is sufficiently area expanding. It is also shown that these systems have exponential decay of correlations for Hölder continuous functions. There are also results on the topological entropy of maps for both classes of maps.



The last paper is about dyadic Diophantine approximation --- the approximation of real numbers by rational numbers of which the denominators are a power of two. We calculate the dimensions of sets with different approximation speeds in the approximation. The paper also contains related results on the approximation of beta-shifts by finite type beta-shifts and the dimension of sets with different speeds of approximation. This is related to the dyadic Diophantine approximation since the the approximation of beta-shifts by finite type beta-shifts can be seen as an approximation of infinite beta-expansion of 1 by finite beta-expansions of 1.



The thesis also contains results on a class of piecewise expanding map on the interval that can be obtained by a projection of maps from the first two articles. These maps are similar to the beta-expansion. The subshift associated to these maps are classified in terms of the orbit of the critical point and some connections between number theoretical properties of the expansion rate and the structure of periodic points are shown. (Less)
Abstract (Swedish)
Popular Abstract in Swedish

Föreliggande avhandling behandlar dynamiken hos styckvist hyperboliska system på begränsade mänder i planet och på intervallet. Avhandlingen är baserad på tre artiklar och därutöver annat stoff.



I de två första artiklarna undersöks två olika klasser av styckvisa affina och hyperboliska avbildningar på begränsade mängder i planet. För båda dessa klasser gäller att tangentrummet kan uppdelas i en kontraherande och en expanderande del och riktningarna för dessa delar är desamma för varje punkt i mängden. Det förs i bevis att för nästan varje parameter så existerar det ett absolutkontinuerligt invariant mått för dessa avbildningar under förutsättningen att de expanderar area... (More)
Popular Abstract in Swedish

Föreliggande avhandling behandlar dynamiken hos styckvist hyperboliska system på begränsade mänder i planet och på intervallet. Avhandlingen är baserad på tre artiklar och därutöver annat stoff.



I de två första artiklarna undersöks två olika klasser av styckvisa affina och hyperboliska avbildningar på begränsade mängder i planet. För båda dessa klasser gäller att tangentrummet kan uppdelas i en kontraherande och en expanderande del och riktningarna för dessa delar är desamma för varje punkt i mängden. Det förs i bevis att för nästan varje parameter så existerar det ett absolutkontinuerligt invariant mått för dessa avbildningar under förutsättningen att de expanderar area tillräckligt mycket. Vidare visas att ifrågavarande avbildningar har exponentiellt avtagande korrelationer för Hölderkontinuerliga funktioner. Avhandlingen innehåller också resultat rörande den topologiska entropin för dylika avbildningar.



Den tredje artikeln, som är författad tillsammans med Jörg Schmeling, behandlar dyadisk Diophantisk approximation; approximation av reella medelst rationella tal med nämnare som är en potens av 2. Dimensionen av olika mängder, där approximationen har en given hastighet, beräknas. Motsvarande gör också för approximationer av beta-skift med beta-skift av ändlig typ.



Avhandlingen innehåller också resultat för en klass av styckvist expanderande avbildningar på intervallet, erhållna genom en projektion av avbildningar från de två första artiklarna. Dessa avbildningar uppvisar likheter med beta-utvecklingen. Skiftrummet till dessa avbildningar klassifieras i termer av egenskaper hos banan av den kritiska punkten och några samband mellan de talteoretiska egenskaperna hos derivatan och de periodiska banornas struktur visas. (Less)
Please use this url to cite or link to this publication:
author
supervisor
opponent
  • Professor Troubetzkoy, Serge, Marseille
organization
publishing date
type
Thesis
publication status
published
subject
keywords
Matematik, Mathematics, Dimension Theory, Hyperbolic Dynamics, Subshift, SRB-measure
pages
134 pages
publisher
Matematikcentrum
defense location
Sal C, Matematikhuset, Sölvegatan 18, Lunds Tekniska Högskola
defense date
2006-12-18 13:15:00
ISBN
978-91-628-6945-8
language
English
LU publication?
yes
id
5eb9840a-fb79-41d1-91a2-7f8846d2406a (old id 547633)
date added to LUP
2016-04-01 15:46:29
date last changed
2018-11-21 20:36:15
@phdthesis{5eb9840a-fb79-41d1-91a2-7f8846d2406a,
  abstract     = {This thesis is about dynamics of piecewise hyperbolic maps on bounded sets in the plane and on the interval. It is based on the following papers:<br/><br>
<br/><br>
T. Persson, A piecewise hyperbolic map with absolutely continuous invariant measure, Dynamical Systems: An International Journal, 21:3 (2006), 363--378<br/><br>
<br/><br>
T. Persson, Absolutely continuous invariant measures for some piecewise hyperbolic affine maps, Preprints in Mathematical Sciences 2005:31, LUTFMA-5066-2005, ISSN 1403-9338<br/><br>
<br/><br>
T. Persson, J. Schmeling, Dyadic Diophantine Approximation and Katok's Horseshoe Approximation,<br/><br>
<br/><br>
Preprints in Mathematical Sciences 2006:3, LUTFMA-5066-2006, ISSN 1403-9338<br/><br>
<br/><br>
The first two papers are about two different classes of piecewise affine hyperbolic maps on bounded subsets of the plane. For both classes the tangent space can be decomposed into one expanding and one contracting subspace and the directions of those subspaces are the same for any point in the manifold. It is shown that, in the sense of parameters, for almost all maps in the two classes there exists an invariant measure that is absolutely continuous with respect to Lebesgue measure, provided that the map is sufficiently area expanding. It is also shown that these systems have exponential decay of correlations for Hölder continuous functions. There are also results on the topological entropy of maps for both classes of maps.<br/><br>
<br/><br>
The last paper is about dyadic Diophantine approximation --- the approximation of real numbers by rational numbers of which the denominators are a power of two. We calculate the dimensions of sets with different approximation speeds in the approximation. The paper also contains related results on the approximation of beta-shifts by finite type beta-shifts and the dimension of sets with different speeds of approximation. This is related to the dyadic Diophantine approximation since the the approximation of beta-shifts by finite type beta-shifts can be seen as an approximation of infinite beta-expansion of 1 by finite beta-expansions of 1.<br/><br>
<br/><br>
The thesis also contains results on a class of piecewise expanding map on the interval that can be obtained by a projection of maps from the first two articles. These maps are similar to the beta-expansion. The subshift associated to these maps are classified in terms of the orbit of the critical point and some connections between number theoretical properties of the expansion rate and the structure of periodic points are shown.},
  author       = {Persson, Tomas},
  isbn         = {978-91-628-6945-8},
  language     = {eng},
  publisher    = {Matematikcentrum},
  school       = {Lund University},
  title        = {Dynamics of Smooth Hyperbolic Systems with Singularities},
  year         = {2006},
}