Rotation number and dynamics of 3-interval piecewise $λ$-affine contractions

TL;DR

This work extends the theory of contracted rotations from 2-interval maps to 3-interval piecewise $\lambda$-affine contractions on $[0,1)$. By constructing an explicit conjugacy $f_{δ,a}\circ φ_{δ,ρ,α}=φ_{δ,ρ,α}\circ R_ρ$ via the function $φ_{δ,ρ,α}$, the authors derive parametric regions $\mathcal{P}_{ρ,α}$ in $(δ,a)$ that realize a prescribed rotation number $ρ$ and control the attractor’s symbolic dynamics through a partition determined by $α$ and $1-ρ$. The main results characterize when the attractor is a Cantor set (for irrational $ρ$) or a union of periodic orbits (for rational $ρ$), and show the attractor’s codes mirror the rotation’s codes, enabling design of maps with specified periodicity and complexity. A reciprocal result confirms that every admissible $(δ,a)$ (outside ghost-fixed-point regions) arises from a chosen $(ρ,α)$, connecting the parameter space directly to the rotation and symbolic dynamics. Overall, this provides a parametric toolkit for constructing 3-interval piecewise contractions with prescribed rotation numbers, attractor types, and complexity, broadening the landscape of explicit, parameter-controlled dynamical behaviors in one-dimensional maps.

Abstract

We consider a family of piecewise contractions admitting a rotation number and defined for every $x\in[0,1)$ by $f(x)=λx + δ+ d θ_a(x) \pmod 1$, where $λ\in(0,1)$, $d\in(0,1-λ)$, $δ\in[0,1]$, $a\in[0,1]$ and $θ_a(x)=1$ if $x\geq a$ and $θ_a(x)=0$ otherwise. In the special case where $a=1$, the family reduces to the well studied ``contracted rotations" $x\mapsto λx + δ\pmod 1$, which are 2-interval piecewise $λ$-affine contractions when $δ\in(1-λ,1)$. Considering $a\in(0,1)$ allows maps with an additional discontinuity, that is, $3$-interval piecewise $λ$-affine contractions. Supposing $λ$ and $d$ fixed, for any $ρ\in(0,1)$ and $α\in[0,1]$, we provide the values of the parameters $δ$ and $a$ for which the corresponding map has rotation number $ρ$, and a symbolic dynamics containing that of the rotation $R_ρ:[0,1)\to[0,1)$ of angle $ρ$ with respect to the partition given by the positions of $1-ρ$ and $α$ in $[0,1)$. This enables in particular to determine the maps that have a given number of periodic orbits of an arbitrary period, or a Cantor set attractor supporting a dynamics of a given complexity.

Figures (7)

• Figure 1: Partition of the set of the parameters $M$ for $\lambda=0.7$ and $d=0.2$ (left), and for $\lambda=0.4$ and $d=0.5$ (right). The dashed regions are the values of parameters $\mathcal{F}_1$ and $\mathcal{F}_2$ for which the map has a fixed point, see \ref{['F1']} and \ref{['F2']}.
• Figure 2: From left to right, an example of map of the type \ref{['MAP2']}, \ref{['MAP3']} and \ref{['MAP']}. All the examples have in common the parameter values $\lambda=0.7$, $d=0.2$ and $\delta=0.5$. They correspond to three values of $(\delta, a)$ taken on the vertical dashed line $\delta=0.5$ of Figure \ref{['FPARAM']} at $a=0.3$ for the leftmost, $a=0.5$ for the middle, and $a=0.8$ for the rightmost example.
• Figure 3: Examples of circle maps with three discontinuities which have a renormalization belonging to the model $M_1$ (left) and $M_3$ (right).
• Figure 4: In green, the set $\mathcal{P}_{\rho,\alpha}$ for $\lambda=0.7, d=0.2$, $\rho=p/q=1/3$ and different values of $\alpha$. Left panel: the parallelogram $\mathcal{P}_{1/3,1/2}$. Right panel: the union of $\mathcal{P}_{1/3,\alpha}$ over all $\alpha\in[0,1]$. The resulting set is made of $2q+1=7$ different regions $\mathcal{P}_{1/3,\alpha}$. We note that $\mathcal{P}_{1/3,1/2}$ is also given by the intersection $\mathcal{P}_{1/3,1/3}\cap\mathcal{P}_{1/3,2/3}$.
• Figure 5: Plot of the $\mathcal{P}_{\rho,\alpha}$ for $\lambda=0.7$, $d=0.2$ and $\rho=1/5, 1/4, 1/3, 2/5$, $1/2, 3/5, 2/3, 3/4, 4/5$ (from left in blue to right in red), with $\alpha \in [0,1-\rho)$ (left panel) $\alpha = 1-\rho$ (middle panel) and $\alpha \in (1-\rho,1]$ (right panel).

Theorems & Definitions (103)

• Remark 2.1
• Definition 2.2
• Theorem 2.3
• Proposition 2.4
• Remark 2.5
• Theorem 2.6
• Corollary 2.7
• proof
• Remark 2.8
• Definition 2.9