Further results for a family of continuous piecewise linear planar maps

TL;DR

This work advances the understanding of the two-dimensional piecewise linear family $F(x,y)=\left(|x| - y + a,\; x - |y| + b\right)$ in the challenging regime $a<0$, where dynamics concentrate on a one-dimensional invariant graph $\Gamma_{a,b}$. It delivers (i) a detailed, continuous characterization of the entropy on $\Gamma$ for $4<c<8$, showing the zero-to-positive transition is not abrupt; (ii) a result guaranteeing a full-measure subset of $\Gamma$ with at most three $\omega$-limit sets, which are periodic when $c\in\mathbb{Q}$; and (iii) a constructive method to obtain sharp rational bounds for the critical $c$-values where entropy changes, including high-precision computations of $\alpha$ and $\beta$ via trapezoidal-map reductions and unimodal dynamics. The paper further connects these dynamics to Markov-graph and rome-based entropy calculations and extends the analytic toolkit with explicit recurrence relations and foldings that reveal the bifurcation structure. These insights sharpen our comprehension of chaotic behavior in planar piecewise-linear maps and illustrate how invariant graphs organize complex dynamics beyond numerical-only observations.

Abstract

We consider the family of piecewise linear maps $F(x,y)=\left(|x| - y + a, x - |y| + b\right),$ where $(a,b)\in \mathbb{R}^2$. In our previous work [10], we presented a comprehensive study of this family. In this paper, we give three new results that complement the ones presented in that reference. All them refer to the most interesting and complicated case, $a<0$. For this case, the dynamics of each map is concentrated in a one-dimensional invariant graph that depend on $b$. In [10], we studied the dynamics of the family on these graphs. In particular, we described whether the topological entropy associated with the map on the graph is positive or zero in terms of the parameter $c=-b/a$. Among the results obtained, we found that there are points of discontinuity of the entropy in the transitions from positive to zero entropy. In this paper, as a first result, we present a detailed explicit analysis of the entropy behavior for the case $4<c<8$, which shows the continuity of this transition from positive to zero entropy. As a second result, we prove that for certain values of the parameter $c$, each invariant graph contains a subset of full Lebesgue measure where there are at most three distinct $ω$-limit sets, which are periodic orbits when $c \in \mathbb{Q}$. Within the framework of the third result, we provide an explicit methodology to obtain accurate rational lower and upper bounds for the values of the parameter $c$ at which the transition from zero to positive entropy occurs.

Figures (9)

• Figure 1: First sets of the partition of $(4,8)$, and exact value or bounds of the entropy according to Proposition \ref{['p:teo1']}. In light blue, the bounding interval in the sets $S_i$ and $U_i$.
• Figure 2: The graph $\Gamma$ for $a=-1$ and $4<b<8$.
• Figure 3: Graphs of the functions $g_\alpha(\lambda)$ (brown), $g_\varphi(\lambda)$ (purple), $g_\delta(\lambda)$ (green), $g_\gamma(\lambda)$ (light blue), and $g_\beta(\lambda)$ (blue) for $\lambda>1$.
• Figure 4: Graphs of $g_\beta(\lambda)$ (blue) and the family of functions $f_n(\lambda)=\lambda^{3n}$ (brown) for $\lambda>1$.
• Figure 5: The graph $\Gamma$ for $a=-1$ and $b\leq -2$.

Theorems & Definitions (18)

• Theorem A
• Theorem B
• Theorem C
• Theorem D
• Theorem 1
• Theorem 2
• Proposition 3
• Lemma 4
• Lemma 5
• Definition 6