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Set-oriented numerical computation of rotation sets
1. | Max Planck Institute for the Physics of Complex Systems, Nöthnitzer Str. 38, 01187 Dresden, Germany |
2. | Leuphana Universität Lüneburg, Institute of Mathematics and its Didactics, Universitätsallee 1, 21335 Lüneburg, Germany |
3. | Friedrich-Schiller-Universität Jena, Institute of Mathematics, Ernst-Abbe-Platz 2, 07743 Jena, Germany |
We establish a set-oriented algorithm for the numerical approximation of the rotation set of homeomorphisms of the two-torus homotopic to the identity. A theoretical background is given by the concept of $\varepsilon$-rotation sets. These are obtained by replacing orbits with $\varepsilon$-pseudo-orbits in the definition of the Misiurewicz-Ziemian rotation set and are shown to converge to the latter as $\varepsilon$ decreases to zero. Based on this result, we prove the convergence of the numerical approximations as precision and iteration time tend to infinity. Further, we provide analytic error estimates for the algorithm under an additional boundedness assumption, which is known to hold in many relevant cases and in particular for non-empty interior rotation sets.
References:
[1] |
S. Addas-Zanata,
Uniform bounds for diffeomorphisms of the torus and a conjecture of boyland, J. Lond. Math. Soc., 91 (2015), 537-553.
doi: 10.1112/jlms/jdu081. |
[2] |
A. Avila, X. -C. Liu and D. Xu, On non-existence of point-wise rotation vectors for minimal toral diffeomorphisms, Preprint, 2016. Google Scholar |
[3] |
F. Béguin, S. Crovisier and F. Le Roux,
Construction of curious minimal uniquely ergodic homeomorphisms on manifolds: The Denjoy-Rees technique, Ann. Sci. Éc. Norm. Supér., 40 (2007), 251-308.
doi: 10.1016/j.ansens.2007.01.001. |
[4] |
P. Boyland, A. de Carvalho and T. Hall,
New rotation sets in a family of torus homeomorphisms, Invent. Math., 204 (2016), 895-937.
doi: 10.1007/s00222-015-0628-2. |
[5] |
P. Davalos,
On annular maps of the torus and sublinear diffusion, Inst. Math. Jussieu, (2016), 1-66.
doi: 10.1017/S1474748016000268. |
[6] |
M. Dellnitz, G. Froyland and O. Junge, The algorithms behind GAIO – set oriented numerical methods for dynamical systems, In Ergodic Theory, Analysis, and Efficient Simulation of Dynamical Systems, Springer, (2001), 145–174, 805–807. |
[7] |
J. Franks,
Realizing rotation vectors for torus homeomorphisms, Trans. Amer. Math. Soc., 311 (1989), 107-115.
doi: 10.1090/S0002-9947-1989-0958891-1. |
[8] |
J. Franks and M. Misiurewicz,
Rotation sets of toral flows, Proc. Amer. Math. Soc., 109 (1990), 243-249.
doi: 10.1090/S0002-9939-1990-1021217-5. |
[9] |
P.-A. Guiheneuf,
How roundoff errors help to compute the rotation set of torus homeomorphisms, Topology App., 193 (2015), 116-139.
doi: 10.1016/j.topol.2015.06.010. |
[10] |
P.-A. Guihéneuf and A. Koropecki,
Stability of the rotation set of area-preserving toral homeomorphisms, Nonlinearity, 30 (2017), 1089-1096.
doi: 10.1088/1361-6544/aa59d9. |
[11] |
T. Jäger,
Elliptic stars in a chaotic night, J. Lond. Math. Soc., 84 (2011), 595-611.
doi: 10.1112/jlms/jdr023. |
[12] |
T. Jäger and A. Passeggi,
On torus homeomorphisms semiconjugate to irrational circle rotations, Ergodic Theory Dynam. Systems, 35 (2015), 2114-2137.
doi: 10.1017/etds.2014.23. |
[13] |
T. Jäger and F. Tal,
Irrational rotation factors for conservative torus homeomorphisms, Ergodic Theory Dynam. Systems, 37 (2017), 1537-1546.
doi: 10.1017/etds.2015.112. |
[14] |
D. B. Johnson,
Finding all the elementary circuits of a directed graph, SIAM J. Comput., 4 (1975), 77-84.
doi: 10.1137/0204007. |
[15] |
A. Kocsard, On the dynamics of minimal homeomorphisms of $\mathbb{T}^2$ which are not pseudo-rotations, Preprint, arXiv: 1611.03784, 2016. Google Scholar |
[16] |
A. Koropecki, A. Passeggi and M. Sambarino, The Franks-Misiurewicz conjecture for extensions of irrational rotations, Preprint, arXiv: 1611.05498, 2016. Google Scholar |
[17] |
A. Koropecki and F. Tal,
Strictly toral dynamics, Invent. Math., 196 (2014), 339-381.
doi: 10.1007/s00222-013-0470-3. |
[18] |
A. Koropecki and F. Tal,
Bounded and unbounded behavior for area-preserving rational pseudo-rotations, Proc. Lond. Math. Soc., 109 (2014), 785-822.
doi: 10.1112/plms/pdu023. |
[19] |
J. Kwapisz,
Every convex polygon with rational vertices is a rotation set, Ergodic Theory Dynam. Systems, 12 (1992), 333-339.
doi: 10.1017/S0143385700006787. |
[20] |
J. Kwapisz,
A toral diffeomorphism with a nonpolygonal rotation set, Nonlinearity, 8 (1995), 461-476.
doi: 10.1088/0951-7715/8/4/001. |
[21] |
P. Le Calvez and S. Addas-Zanata, Rational mode locking for homeomorphisms of the 2-torus, Preprint, arXiv: 1508.02597, 2015. Google Scholar |
[22] |
P. Le Calvez and F. Tal, Forcing theory for transverse trajectories of surface homeomorphisms, Preprint, arXiv: 1503.09127, 2015. Google Scholar |
[23] |
P. Leboeuf, J. Kurchan, M. Feingold and D.P. Arovas,
Phase-space localization: topological aspects of quantum chaos, Phys. Rev. Lett., 65 (1990), 3076-3079.
doi: 10.1103/PhysRevLett.65.3076. |
[24] |
J. Llibre and R. S. MacKay,
Rotation vectors and entropy for homeomorphisms of the torus isotopic to the identity, Ergodic Theory Dynam. Systems, 11 (1991), 115-128.
doi: 10.1017/S0143385700006040. |
[25] |
M. Misiurewicz and K. Ziemian,
Rotation sets for maps of tori, J. Lond. Math. Soc., 40 (1989), 490-506.
doi: 10.1112/jlms/s2-40.3.490. |
[26] |
A. Passeggi,
Rational polygons as rotation sets of generic torus homeomorphisms of the two torus, J. Lond. Math. Soc., 89 (2014), 235-254.
doi: 10.1112/jlms/jdt040. |
[27] |
H. Poincaré, Mémoire sur les courbes définies par une équation différentielle, J. Math. Pure. Appl., Série IV, 1 (1885), 167-244. Google Scholar |
show all references
References:
[1] |
S. Addas-Zanata,
Uniform bounds for diffeomorphisms of the torus and a conjecture of boyland, J. Lond. Math. Soc., 91 (2015), 537-553.
doi: 10.1112/jlms/jdu081. |
[2] |
A. Avila, X. -C. Liu and D. Xu, On non-existence of point-wise rotation vectors for minimal toral diffeomorphisms, Preprint, 2016. Google Scholar |
[3] |
F. Béguin, S. Crovisier and F. Le Roux,
Construction of curious minimal uniquely ergodic homeomorphisms on manifolds: The Denjoy-Rees technique, Ann. Sci. Éc. Norm. Supér., 40 (2007), 251-308.
doi: 10.1016/j.ansens.2007.01.001. |
[4] |
P. Boyland, A. de Carvalho and T. Hall,
New rotation sets in a family of torus homeomorphisms, Invent. Math., 204 (2016), 895-937.
doi: 10.1007/s00222-015-0628-2. |
[5] |
P. Davalos,
On annular maps of the torus and sublinear diffusion, Inst. Math. Jussieu, (2016), 1-66.
doi: 10.1017/S1474748016000268. |
[6] |
M. Dellnitz, G. Froyland and O. Junge, The algorithms behind GAIO – set oriented numerical methods for dynamical systems, In Ergodic Theory, Analysis, and Efficient Simulation of Dynamical Systems, Springer, (2001), 145–174, 805–807. |
[7] |
J. Franks,
Realizing rotation vectors for torus homeomorphisms, Trans. Amer. Math. Soc., 311 (1989), 107-115.
doi: 10.1090/S0002-9947-1989-0958891-1. |
[8] |
J. Franks and M. Misiurewicz,
Rotation sets of toral flows, Proc. Amer. Math. Soc., 109 (1990), 243-249.
doi: 10.1090/S0002-9939-1990-1021217-5. |
[9] |
P.-A. Guiheneuf,
How roundoff errors help to compute the rotation set of torus homeomorphisms, Topology App., 193 (2015), 116-139.
doi: 10.1016/j.topol.2015.06.010. |
[10] |
P.-A. Guihéneuf and A. Koropecki,
Stability of the rotation set of area-preserving toral homeomorphisms, Nonlinearity, 30 (2017), 1089-1096.
doi: 10.1088/1361-6544/aa59d9. |
[11] |
T. Jäger,
Elliptic stars in a chaotic night, J. Lond. Math. Soc., 84 (2011), 595-611.
doi: 10.1112/jlms/jdr023. |
[12] |
T. Jäger and A. Passeggi,
On torus homeomorphisms semiconjugate to irrational circle rotations, Ergodic Theory Dynam. Systems, 35 (2015), 2114-2137.
doi: 10.1017/etds.2014.23. |
[13] |
T. Jäger and F. Tal,
Irrational rotation factors for conservative torus homeomorphisms, Ergodic Theory Dynam. Systems, 37 (2017), 1537-1546.
doi: 10.1017/etds.2015.112. |
[14] |
D. B. Johnson,
Finding all the elementary circuits of a directed graph, SIAM J. Comput., 4 (1975), 77-84.
doi: 10.1137/0204007. |
[15] |
A. Kocsard, On the dynamics of minimal homeomorphisms of $\mathbb{T}^2$ which are not pseudo-rotations, Preprint, arXiv: 1611.03784, 2016. Google Scholar |
[16] |
A. Koropecki, A. Passeggi and M. Sambarino, The Franks-Misiurewicz conjecture for extensions of irrational rotations, Preprint, arXiv: 1611.05498, 2016. Google Scholar |
[17] |
A. Koropecki and F. Tal,
Strictly toral dynamics, Invent. Math., 196 (2014), 339-381.
doi: 10.1007/s00222-013-0470-3. |
[18] |
A. Koropecki and F. Tal,
Bounded and unbounded behavior for area-preserving rational pseudo-rotations, Proc. Lond. Math. Soc., 109 (2014), 785-822.
doi: 10.1112/plms/pdu023. |
[19] |
J. Kwapisz,
Every convex polygon with rational vertices is a rotation set, Ergodic Theory Dynam. Systems, 12 (1992), 333-339.
doi: 10.1017/S0143385700006787. |
[20] |
J. Kwapisz,
A toral diffeomorphism with a nonpolygonal rotation set, Nonlinearity, 8 (1995), 461-476.
doi: 10.1088/0951-7715/8/4/001. |
[21] |
P. Le Calvez and S. Addas-Zanata, Rational mode locking for homeomorphisms of the 2-torus, Preprint, arXiv: 1508.02597, 2015. Google Scholar |
[22] |
P. Le Calvez and F. Tal, Forcing theory for transverse trajectories of surface homeomorphisms, Preprint, arXiv: 1503.09127, 2015. Google Scholar |
[23] |
P. Leboeuf, J. Kurchan, M. Feingold and D.P. Arovas,
Phase-space localization: topological aspects of quantum chaos, Phys. Rev. Lett., 65 (1990), 3076-3079.
doi: 10.1103/PhysRevLett.65.3076. |
[24] |
J. Llibre and R. S. MacKay,
Rotation vectors and entropy for homeomorphisms of the torus isotopic to the identity, Ergodic Theory Dynam. Systems, 11 (1991), 115-128.
doi: 10.1017/S0143385700006040. |
[25] |
M. Misiurewicz and K. Ziemian,
Rotation sets for maps of tori, J. Lond. Math. Soc., 40 (1989), 490-506.
doi: 10.1112/jlms/s2-40.3.490. |
[26] |
A. Passeggi,
Rational polygons as rotation sets of generic torus homeomorphisms of the two torus, J. Lond. Math. Soc., 89 (2014), 235-254.
doi: 10.1112/jlms/jdt040. |
[27] |
H. Poincaré, Mémoire sur les courbes définies par une équation différentielle, J. Math. Pure. Appl., Série IV, 1 (1885), 167-244. Google Scholar |










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