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Set-oriented numerical computation of rotation sets

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  • 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.

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

    Citation:

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

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

    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)

    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

    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)

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

    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

    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)

    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)

    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 $i=35$

    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)

    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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