Numerical analysis for a discontinuous rotation of the torus

Abstract

In this paper, we study a class of piecewise rotations on the square. While few theoretical results are known about them, we numerically compute box-counting dimensions, correlation dimensions and complexity of the symbolic language produced by the system. Our results seem to confirm a conjecture that the fractal dimension of the exceptional set is two, as well as indicate that the dynamics on it is not ergodic. We also explore a relationship between the piecewise rotations and discretized rotations on lattices Z 2n . © 2003 American Institute of Physics. ͓DOI: 10.1063/1.1572411͔It is well known that simply defined dynamical systems can exhibit very complex behavior. In this paper we will consider a family of piecewise isometries, namely of discontinuous rotations on the two-torus. These systems are not ergodic; their various invariant sets display a rich and complicated geometry and support dynamics with nontrivial statistical properties. Most of the characteristics of these systems are not completely understood. Basic questions such as the distribution of periodic points and the coding of the trajectories on the closure of the discontinuity lines, are still open. We will present a careful numerical investigation of three features of these maps. First, we compute the fractal dimension of the exceptional set "i.e., the complement of the collection of ''periodic islands''…. There is evidence that this exceptional set may have a positive Lebesgue measure. Second, we investigate the complexity of the orbits on the exceptional set; we found several "piecewise… polynomial behaviors of the complexity function, some referring to a substitution system. Third, we show a correspondence between discontinuous rotations of the torus and discretized rotations on a lattice. This correspondence may help in explaining the differences in the growth of the complexity function.

The unique ergodic measure of the symmetric piecewise toral isometry of rotation angle θ=kπ/5 is the Hausdorff measure of its singular set

“…The emergence of an invariant fractal due to the discontinuity of F is well documented (Chua and Lin 1988, Davis 1995, Ashwin 1996, Kocarev et al 1996, Kahng 2002, 2004a, Bruin 2004, since its discovery (Anderson and Galway 1977). In our system (Definition 1), the discontinuity forF F andF F À1 is respectively fðx, 0Þ : 0 x < 1g and fð0, yÞ : 0 y < 1g.…”

“…The periodicity and the singularity structure of F are being studied both analytically (Kahng 2000, 2002, 2004a, Alder et al 2001, Koupstov et al 2002 and numerically (Chua and Lin 1988, 1990a,b, Ogorzalek 1992, Davis 1995, Ashwin 1996, Kocarev et al 1996, Bruin 2004. This is the oldest example (Anderson and Galway 1977) of piecewise elliptic and/or piecewise isometric dynamics, which is being studied in conjunction with various fields such as billiards (Boldrigini et al 1976, Gutkin 1996, Gutkin et al 1997, dual billiards (Gutkin andSimanyi 1992, Tabachnikov 1995a,b), Hamiltonian dynamics (MacKay and Meiss 1987, Lowenstein 1995, Lowenstein et al 1997, Lowenstein and Vivaldi 1998, Kahng 2000, and signal processing through lossless digital filters (Chua and Lin 1988, 1990a,b, Ogorzalek 1992, Lin and Chua 1993, Davis 1995, Ashwin 1996, Kocarev et al 1996.…”

Pseudochaotic dynamics near global periodicity

“…On the other side, the irrational rotation angle cases display even more complex structure and possess more difficulties to calculate dimension exactly, which leaves numerical method the best, if not the only, way. Numerical investigation [4] supported the conjecture that irrational rotation angle cases have dimension 2. Also, it was conjectured and numerically supported by Ashwin [2,3] that exceptional set has positive Lebesgue measure.…”

“…Since then, the dynamics of the piecewise linear maps has been extensively studied and is becoming an important branch of dynamical system theory. Besides lossless digital filters, piecewise linear maps have appeared in study of dual billiards dynamics [16,[33][34][35][36], polygonal exchange transformations [4,13], round-off errors in linear systems [23,26,29,30,27] and Ergodic theory [1,5,19,20]. Recently it is considered as an example of pseudo-chaos [31,12].…”

Return maps in cyclotomic piecewise similarities

“…This is an example of a class of maps studied in electrical engineering literature for example in [7][8][9][10][11]. A particular case of this class corresponding to a rotation by /4 was extensively studied as an example in [2].…”