On the Piecewise Linear Perturbations of the Doubling Map

TL;DR

The paper studies the piecewise linear perturbations of the circle doubling map (PLPDM), defined by $f_{a,b}(x)=\bigl(2x+a+\dfrac{b}{2}S(x)\bigr)//1$ with a non-analytic, piecewise-linear perturbation $S(x)$. It develops a framework of hyperbolic components (tongues and eyes) in the parameter space and proves the uniqueness of attracting cycles within the hyperbolic set, along with a semiconjugacy to the doubling map that enables a notion of Type for tongues. The authors establish the existence of tongues for all types, providing explicit constructions for periods 3 and 4 and a general algorithm for arbitrary periods, and show that non-single-minus itineraries correspond to eyes. They also show that outside $\overline{\mathcal{H}}$, PLPDM maps are conjugate to the doubling map, while near the boundaries, neutral cycles arise at the break points, signaling bifurcations. The work highlights qualitative differences between PLPDM and analytic DSM families, including the emergence of eyes and a broader realization of combinatorial types, and raises several open questions about post-critical dynamics and uniformization of hyperbolic components.

Abstract

Inspired by the 2007 work by M.~Misiurewicz and A.~Rodrigues [Double Standard Maps, M. Misiurewicz, A. Rodrigues, Communications in Mathematical Physics], we consider a family of circle maps that are perturbations of the doubling map on the circle by a piecewise linear map. We call this the \textit{piecewise linear perturbation of the doubling map} (PLPDM) and it is given by the formula, $f_{a,b}(x)= \displaystyle \bigl(2x+a+\dfrac{b}{2} S(x) \bigr) // 1 \quad {\text {for }} x, a, b \in [0,1] $, where $y // 1$ means $y \mod 1$ (or simply, the fractional part of $y$) and $S(x)$ is the piecewise linear approximation of $\sin 2π(x-1/4)$. The map $S(x)$ is called the straight sine map. Define the hyperbolic set, $\mathcal{H}= \{ (a,b) \in \mathbb{R}/\mathbb{Z} \times [0,1] : f_{a,b} \text{ has an attracting cycle} \}$. Tongues are defined as the components of $\mathcal{H}$ that touch the ceiling $\{b=1\}$ in a non degenerate interval. Any other component is referred to as an Eye. We show the uniqueness of the attracting cycle of $f_{a,b}$ for $(a,b) \in \mathcal{H}$. We then define \textit{type} and prove the existence of the tongues of all types. We also show how combinatorics of the attracting orbit determines if the component is a tongue or an eye. We show that $f_{a,b}$ is conjugate to the doubling map if $(a,b) \notin \overline{\mathcal{H}}$. Some experimental proof of the existence of eyes in the parameter space corresponding to different combinatorics will be shown.

Figures (11)

• Figure 1: Tongues and Eyes in PLPDM (By A. Dezotti)
• Figure 2: Schematic diagram of semiconjugacy
• Figure 3: Cobweb diagrams for three different cases.
• Figure 4: $a=0.964, b=0.988, A^+=.47$
• Figure 5: At height $b=0.999$, the horizontal span of period $5$ tongue is from $a\sim0.712957676959782$ (left figure) to $a\sim0.71367603$ (right figure)

Theorems & Definitions (60)

• Definition 2.1: Itinerary
• Remark 1
• Definition 2.2: Tongue and Eye
• Definition 2.3: Neutral Cycle
• Theorem 2.4
• Theorem 2.5
• Lemma 3.1
• Lemma 3.2
• proof
• Theorem 3.3