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Piecewise continuous and monotonic maps on the interval

Kleyber Cunha, Marcio Gouveia, Paulo Santana

TL;DR

This work addresses the dynamics of piecewise continuous and monotonic maps on an interval, where finitely many discontinuities and turning points yield phenomena absent in continuous maps. It develops a unifying framework based on variants $\mathcal{E}(f)$, the orbit-structure $O((x))$, and closed structures $[[x]]$, and analyzes periodic, continuous periodic, and closed orbital structures. Key contributions include propagation of stability along closed structures under limited discontinuities, a detailed classification of trapped, free, and critical orbits, and a coding-based description of dynamics away from discontinuities, along with the notion of attraction sets $A([x],f)$ for these structures. Together, these results extend classic one-dimensional dynamics to discontinuous, piecewise settings and provide tools applicable to Lorenz-type maps and other systems with piecewise behavior.

Abstract

Let $f$ be a piecewise continuous and monotonic map on the interval with at most finitely many discontinuities and turning points. In this paper we study properties about this class of maps and show its main difference from the continuous case. We define and study the notion of closed structure, which can be seen as an generalization of periodic orbit. We also study the periodic orbits that are away from the discontinuities of $f$, extending the notion of trapped and free orbits.

Piecewise continuous and monotonic maps on the interval

TL;DR

This work addresses the dynamics of piecewise continuous and monotonic maps on an interval, where finitely many discontinuities and turning points yield phenomena absent in continuous maps. It develops a unifying framework based on variants , the orbit-structure , and closed structures , and analyzes periodic, continuous periodic, and closed orbital structures. Key contributions include propagation of stability along closed structures under limited discontinuities, a detailed classification of trapped, free, and critical orbits, and a coding-based description of dynamics away from discontinuities, along with the notion of attraction sets for these structures. Together, these results extend classic one-dimensional dynamics to discontinuous, piecewise settings and provide tools applicable to Lorenz-type maps and other systems with piecewise behavior.

Abstract

Let be a piecewise continuous and monotonic map on the interval with at most finitely many discontinuities and turning points. In this paper we study properties about this class of maps and show its main difference from the continuous case. We define and study the notion of closed structure, which can be seen as an generalization of periodic orbit. We also study the periodic orbits that are away from the discontinuities of , extending the notion of trapped and free orbits.
Paper Structure (4 sections, 15 theorems, 60 equations, 11 figures)

This paper contains 4 sections, 15 theorems, 60 equations, 11 figures.

Table of Contents

  1. Introduction
  2. Definitions and basic properties
  3. Periodic orbits and closed structures
  4. Critical, trapped and free continuous periodic orbits

Key Result

Proposition 1

Let $f\in P([a,b])$. Then $f^{-1}(y)$ is finite, for every $y\in[a,b]$.

Figures (11)

  • Figure 1: Illustration of a well behaved piecewise continuous map.
  • Figure 2: An illustration of the graphs of $f$ and $f^2$, given by the thicker lines.
  • Figure 3: An illustration of $O((x))$.
  • Figure 4: An illustration of two distinct periodic orbits with intersection.
  • Figure 5: An illustration of a closed structure $[[x]]$. Observe that $[[x]]$ has a periodic orbit $[x]=\{x,\dots,x_8\}$ of period $9$; a periodic orbit $[y]=\{y,y_1,y_2,y_3\}$ of period $4$ and a fixed point $z$.
  • ...and 6 more figures

Theorems & Definitions (39)

  • Proposition 1
  • proof
  • Proposition 2
  • proof
  • Proposition 3
  • proof
  • Definition 1
  • Remark 1
  • Definition 2
  • Remark 2
  • ...and 29 more