Set-oriented numerical computation of rotation sets
Publikation: Beiträge in Zeitschriften › Zeitschriftenaufsätze › Forschung › begutachtet
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Set-oriented numerical computation of rotation sets. / Polotzek, Katja; Padberg-Gehle, Kathrin; Oertel-Jäger, Tobias.
in: Journal of Computational Dynamics, Jahrgang 4, Nr. 1, 01.11.2017, S. 119-141.Publikation: Beiträge in Zeitschriften › Zeitschriftenaufsätze › Forschung › begutachtet
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TY - JOUR
T1 - Set-oriented numerical computation of rotation sets
AU - Polotzek, Katja
AU - Padberg-Gehle, Kathrin
AU - Oertel-Jäger, Tobias
N1 - Publisher Copyright: © American Institute of Mathematical Sciences.
PY - 2017/11/1
Y1 - 2017/11/1
N2 - We establish a set-oriented algorithm for the numerical approximationof the rotation set of homeomorphisms of the two-torus homotopic tothe identity. A theoretical background is given by the concept of ε-rotationsets. These are obtained by replacing orbits with ε-pseudo-orbits in the definitionof the Misiurewicz-Ziemian rotation set and are shown to converge to thelatter as ε decreases to zero. Based on this result, we prove the convergenceof the numerical approximations as precision and iteration time tend to infinity.Further, we provide analytic error estimates for the algorithm under anadditional boundedness assumption, which is known to hold in many relevantcases and in particular for non-empty interior rotation sets.
AB - We establish a set-oriented algorithm for the numerical approximationof the rotation set of homeomorphisms of the two-torus homotopic tothe identity. A theoretical background is given by the concept of ε-rotationsets. These are obtained by replacing orbits with ε-pseudo-orbits in the definitionof the Misiurewicz-Ziemian rotation set and are shown to converge to thelatter as ε decreases to zero. Based on this result, we prove the convergenceof the numerical approximations as precision and iteration time tend to infinity.Further, we provide analytic error estimates for the algorithm under anadditional boundedness assumption, which is known to hold in many relevantcases and in particular for non-empty interior rotation sets.
KW - Mathematics
KW - Rotation Theory
KW - rotation sets
KW - pseudo-orbits
KW - set-oriented numerics
UR - http://www.scopus.com/inward/record.url?scp=85040731579&partnerID=8YFLogxK
UR - https://www.mendeley.com/catalogue/baa4c899-3756-3249-b150-8a6f1cbd87c0/
U2 - 10.3934/jcd.2017004
DO - 10.3934/jcd.2017004
M3 - Journal articles
VL - 4
SP - 119
EP - 141
JO - Journal of Computational Dynamics
JF - Journal of Computational Dynamics
SN - 2158-2491
IS - 1
ER -