Piecewise linear circle maps and conjugation to rigid rational rotations

TL;DR

This work characterizes when piecewise-linear circle maps with rational rotation numbers are conjugate to rigid rational rotations, showing that conjugacy by a PWL map is equivalent to all break points being periodic and obeying a jump condition on slopes. It further demonstrates that, in monotone one-parameter families, the presence of a parameter where the map is conjugate to a rigid rotation eliminates mode-locking for that rotation number and induces a linear scaling of the rotation number near that parameter. The paper develops a detailed framework linking break-point dynamics, absolutely continuous invariant measures, and growth properties of iterates, and illustrates the theory with Herman's classic two-breakpoint model and a four-breakpoint map arising from refraction in a periodic medium. These results connect rational-rotation behavior in PWL maps to classical irrational-rotation theory, and reveal pinching and codimension phenomena in parameter spaces relevant to mode-locking and rigidity.

Abstract

Criteria for piecewise linear circle homeomorphisms to be conjugate to a rigid rotation, $x\to x+ω~({\rm mod}~1)$, with rational rotation number $ω$ are given. The consequences of the existence of such maps in families of maps is considered and the results are illustrated using two examples: Herman's classic family of piecewise linear maps with two linear components, and a map derived from geometric optics which has four components. These results show how results for piecewise smooth circle homeomorphisms with irrational rotation numbers have natural correspondences with the case of rational rotation numbers for piecewise linear maps. In natural families of maps the existence of a parameter value at which the map is conjugate to a rigid rotation implies linear scaling of the rotation number in a neighbourhood of the critical parameter value and no mode-locked intervals, in contrast to the behaviour of generic families of circle maps.

Figures (5)

• Figure 1: Plot of the rotation number $\rho$ (vertical axis) of (\ref{['eq:GMMex']}) against the parameter $-\beta$. The rotation number is calculated using 100000 iterates of the map at 100 equidistant values of the parameter with $\alpha=2$ and $1.122\le \beta < 1.131$. Part of the mode locked region with rotation number $\frac{5}{6}$ is visible on the left of the figure. The parameter $-\beta$ is used in the plot so that the rotation number increases with parameter as is the convention in the theoretical part of this paper.
• Figure 2: Sketch of a lift $f_{\alpha ,\beta}$ and the iterate $f_{\alpha ,\beta}^5$ shifted by 4 using (\ref{['eq:GMMex']}) with $\alpha =2$ and $\beta=(\sqrt{5}-1)-\mu$, $\mu=0.02$ (cf. Figure \ref{['fig:ourfig']}a). As discussed in section \ref{['sect:GMMex']}, $f_{\alpha ,\beta}$ has four break points and if $\mu =0$ these lie on the same orbit of period five which is indicated by the circles on the diagonal. The intervals $L_i$ are centred on these (the point which is not a break point is in $L_4$, and contain all the break points of $F_{\alpha ,\beta}^5$. Outside these regions the map has slope equal to one (in general the slope would be close to one) Note that $L_1$ is centred on $x=0$ and so straddles the periodic window presented here.
• Figure 3: Rotation number of (\ref{['def:R']}) as a function of the parameter $\mu$. The rotation number is calculated using 100000 iterates of the map at 1000 equidistant values of the parameter between $-0.2$ and $0.2$.
• Figure 4: Rotation number (vertical axis) of (\ref{['eq:GMMex']}) against the parameter $\mu$. The rotation number is calculated using 100000 iterates of the map at 1000 equidistant values of the parameter $\mu$. (a) $\alpha=2$, $\beta=(\sqrt{5}-1)+\mu$, $-0.1<\mu <0.1$; (b) $m=\frac{\alpha}{\beta}=\sqrt{10}$, $\alpha =\alpha_0+\mu$, $\alpha_0= 3.403$, $-0.1<\mu<0.1$.
• Figure 5: (a) Schematic view of a mode locked region in two parameter space showing pinching at the point at which the map is conjugate to a rigid rational rotation. (b) Linear approximation of the equivalent regions with $\frac{p}{q}=\frac{1}{2}$ for the example (\ref{['ex:Hmod']}) with $0<\lambda <1$. The rotation number is $\frac{1}{2}$ in the wedge-shaped regions labelled $\frac{1}{2}$.

Theorems & Definitions (21)

• Theorem 2.1
• Theorem 2.2
• Theorem 2.3
• Theorem 2.4
• Definition 2.1
• Theorem 2.5
• Theorem 2.6
• Lemma 3.1
• Corollary 5.1
• Lemma 5.2