Set-oriented numerical computation of rotation sets

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June 2017, 4(1&2): 119-141. doi: 10.3934/jcd.2017004

Set-oriented numerical computation of rotation sets

Katja Polotzek ^1, , Kathrin Padberg-Gehle ^2, and Tobias Jäger ^3,

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

Published November 2017

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.

Keywords: Rotation theory, rotation sets, pseudo-orbits, set-oriented numerics.

Mathematics Subject Classification: Primary: 65P99; Secondary: 37M25, 37E45.

Citation: Katja Polotzek, Kathrin Padberg-Gehle, Tobias Jäger. Set-oriented numerical computation of rotation sets. Journal of Computational Dynamics, 2017, 4 (1&2) : 119-141. doi: 10.3934/jcd.2017004

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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[5]	P. Davalos, On annular maps of the torus and sublinear diffusion, Inst. Math. Jussieu, (2016), 1-66. doi: 10.1017/S1474748016000268. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[11]	T. Jäger, Elliptic stars in a chaotic night, J. Lond. Math. Soc., 84 (2011), 595-611. doi: 10.1112/jlms/jdr023. Google Scholar
[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. Google Scholar
[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. Google Scholar
[14]	D. B. Johnson, Finding all the elementary circuits of a directed graph, SIAM J. Comput., 4 (1975), 77-84. doi: 10.1137/0204007. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[20]	J. Kwapisz, A toral diffeomorphism with a nonpolygonal rotation set, Nonlinearity, 8 (1995), 461-476. doi: 10.1088/0951-7715/8/4/001. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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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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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[5]	P. Davalos, On annular maps of the torus and sublinear diffusion, Inst. Math. Jussieu, (2016), 1-66. doi: 10.1017/S1474748016000268. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[11]	T. Jäger, Elliptic stars in a chaotic night, J. Lond. Math. Soc., 84 (2011), 595-611. doi: 10.1112/jlms/jdr023. Google Scholar
[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. Google Scholar
[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. Google Scholar
[14]	D. B. Johnson, Finding all the elementary circuits of a directed graph, SIAM J. Comput., 4 (1975), 77-84. doi: 10.1137/0204007. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[20]	J. Kwapisz, A toral diffeomorphism with a nonpolygonal rotation set, Nonlinearity, 8 (1995), 461-476. doi: 10.1088/0951-7715/8/4/001. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[27]	H. Poincaré, Mémoire sur les courbes définies par une équation différentielle, J. Math. Pure. Appl., Série IV, 1 (1885), 167-244. Google Scholar

Figure 4.1. Approximation of $\varrho{F_{1,1}}$ by a direct approach, $80$ (left) and $2500$ (right) iterations, grid range $0.001$ each

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Figure 4.2. Box image of a box $B$ in the box covering $\mathcal{B}_k$, for one test point, exemplarily

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Figure 5.1. Approximations $Q^*_{k,n}$ for the rotation set of the map $F_{1,1}$ with $k=8$ and $n = 1,2,5,10,25,50,100,200$ (from top left to bottom right)

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Figure 5.2. Zoom on top left area of the approximations $Q_{8,100}^\ast$ (left) and $Q_{8,200}^\ast$ (right) for the rotation set of the map $F_{1,1}$. The shaded area is the $2\sqrt{2}/n$-neighbourhood of these sets, which is a superset of $\rho(F_{1,1})$ by Lemma 19

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Figure 5.3. Approximations $Q_{k,n}^*$ for the rotation sets of the maps $F_{^1{/_2},^1{/_2}}, F_{1,{^1{/_4}}}, F_{^{3 }{/_{5}},^{3 }{/_{5}}}$ and $F_{{^3{/_4}},1}$ (with $k=50, 16, 50,45$ and $n=130,140,100,80$ from top left to bottom right)

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Figure 5.4. Approximation $Q_{60,130}^\ast$ for the rotation set of the map $G$

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Figure 5.5. Approximations $Q_{k,n}^*$ for the rotation sets of the perturbed maps $\bar{F}_{^1{/_2},^1{/_2}}, \bar{G}, \bar{F}_{1,{^1{/_4}}}, \bar{F}_{^{3 }{/_{5}},^{3 }{/_{5}}}$, $\bar{F}_{{^3{/_4}},1}$ and $\bar{F}_{1,1}$ (from top left to bottom right) according to Table 1

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Figure 5.6. Approximations $Q^*_{k,n}$ of the rotation set of $F_{0.873,0.873}$ with $n=50$ and $k=15,16,20,25,30,40,50,80$ (from top left to bottom right)

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Figure 5.7. Approximations $Q^*_{k,n}$ of the rotation set of the map $F_{0.873,0.873}$ with $n=100$ and $k=15,20,40,50$ (from top left to bottom right)

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Figure 5.8. Approximations of the rotation sets of $F_{\alpha_i,\beta_i}$ taken from a series with parameters $\alpha_i=\beta_i=0.02\cdot i$ and an adapted choice for the iteration time and grid sizes $n(i)=k(i)=110-i$. Pictures are shown for $i=30,35,36,37$ (from left to right), indicating a mode-locking region from $i=30$ to $i=35$

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Figure 5.9. A closer look at the parameter region considered in Figure 5.8. Rotation sets were approximated for parameters $\alpha_j=\beta_j=0.7+0.01\cdot j$, with $j=0, \ldots , 7$ and $n=75$ and $k=90$ fixed. Note the difference between the third picture in the first line and the third in Figure 5.8, which both correspond to parameters $\alpha=\beta=0.72$ (but different precisions)

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Table 1. Parameter values for the approximations shown in Figure 5.5

	$\bar{F}_{\frac{1}{2},\frac{1}{2}}$	$\bar{G}$	$\bar{F}_{1,\frac{1}{4}}$	$\bar{F}_{\frac{3}{5},\frac{3}{5}}$	$\bar{F}_{\frac{3}{4},1}$	$\bar{F}_{1,1}$
$k$	$50$	$60$	$16$	$50$	$45$	$8$
$n$	$130$	$130$	$140$	$100$	$80$	$100$
$r_1$	$0.012$	$0.008$	$0.012$	$0.01$	$0.002$	$0.022$
$r_2$	$0.014$	$0.001$	$0.002$	$0.011$	$0.013$	$0.015$

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	$\bar{F}_{\frac{1}{2},\frac{1}{2}}$	$\bar{G}$	$\bar{F}_{1,\frac{1}{4}}$	$\bar{F}_{\frac{3}{5},\frac{3}{5}}$	$\bar{F}_{\frac{3}{4},1}$	$\bar{F}_{1,1}$
$k$	$50$	$60$	$16$	$50$	$45$	$8$
$n$	$130$	$130$	$140$	$100$	$80$	$100$
$r_1$	$0.012$	$0.008$	$0.012$	$0.01$	$0.002$	$0.022$
$r_2$	$0.014$	$0.001$	$0.002$	$0.011$	$0.013$	$0.015$

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