Dynamics of a family of piecewise-linear area-preserving plane maps I. Rational rotation numbers

Abstract

This paper studies the behavior under iteration of the maps T ab (x, y) = (F ab (x)−y, x) of the plane R 2 , in which F ab (x) = ax if x ≥ 0 and bx if x < 0. The orbits under iteration correspond to solutions of the nonlinear difference equationThis family of piecewise-linear maps has the parameter space (a, b) ∈ R 2 . These maps are area-preserving homeomorphisms of R 2 that map rays from the origin into rays from the origin. The action on rays gives an auxiliary map S ab : S 1 → S 1 of the circle, which has a well-defined rotation number. This paper characterizes the possible dynamics under iteration of T ab when the auxiliary map S ab has rational rotation number. It characterizes cases where the map T ab is a periodic map.

Bifurcation Patterns in Homogeneous Area-Preserving Piecewise-Linear Maps

“…Also, if the rotation number is rational, let us say ρ = m/n with m and n coprime, then the map S has a period-n orbit at least, and all the existing periodic orbits are period-n orbits. The dynamics of the map S in case of rational rotation number is characterized in [10] as follows.…”

“…Here, we are not directly interested in the border-collision bifurcation, and following [10]- [12], we deal with the homogeneous case m = 0, relegating the non-homogeneous case m = 1 to future work. Moreover, we will pay attention only to the non-generic but important case of area-preserving maps, that is we will assume D + = D − = 1, so that the system to be studied becomes, (5) x…”

“…Family (5) has been already studied in a series of papers by Lagarias and Rains, see [10], [11] and [12], where many properties of periodic behavior were obtained, see for instance Theorem 2.4 below. Also, parametrical regions with fixed rational rotation number (for certain rationals) were reported but not completely described.…”

Arithmetic exponents in piecewise-affine planar maps

“…If d = 1, then for any choice of f the map F is area-preserving (see section 4). The literature devoted to maps of this type is substantial [7,4,3,1,16,17,18].…”

Dynamics of a family of piecewise-linear area-preserving plane maps III. Cantor set spectra

“…As in parts I and II [12,13], we study the behavior under iteration of the two parameter family of piecewise-linear homeomorphisms of R 2 given by b if x , 0:…”