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Forcing among exact patterns of triods

Sourav Bhattacharya

TL;DR

The article provides a complete characterization of topologically exact patterns on triods by introducing rotation-based classifications (slow, fast, ternary) and proving that exactness corresponds to the absence of block structure in patterns. It develops a robust framework of blocks, loops, and $P$-linear maps to analyze forcing among patterns, and derives three explicit, perturbation-stable orderings on natural numbers that capture forcing within each rotation-number class. These results extend Blokh–Misiurewicz-style rotation theory to triods, offering precise predictions for coexistence of exact patterns and advancing the understanding of combinatorial dynamics on branched one-dimensional spaces.

Abstract

We obtain a complete characterization of \emph{topologically exact patterns} on \emph{triods}. Based on their \emph{rotation number} $ρ$, these \emph{exact patterns} are grouped into three classes: \emph{slow} ($ρ< \frac{1}{3}$), \emph{fast} ($ρ> \frac{1}{3}$) and \emph{ternary} ($ρ= \frac{1}{3}$). For each category, we derive a \emph{linear ordering} of the set of natural numbers, $\mathbb{N}$ that captures \emph{forcing} between the \emph{patterns}. We also show that each of these orderings is \emph{stable} under perturbations.

Forcing among exact patterns of triods

TL;DR

The article provides a complete characterization of topologically exact patterns on triods by introducing rotation-based classifications (slow, fast, ternary) and proving that exactness corresponds to the absence of block structure in patterns. It develops a robust framework of blocks, loops, and -linear maps to analyze forcing among patterns, and derives three explicit, perturbation-stable orderings on natural numbers that capture forcing within each rotation-number class. These results extend Blokh–Misiurewicz-style rotation theory to triods, offering precise predictions for coexistence of exact patterns and advancing the understanding of combinatorial dynamics on branched one-dimensional spaces.

Abstract

We obtain a complete characterization of \emph{topologically exact patterns} on \emph{triods}. Based on their \emph{rotation number} , these \emph{exact patterns} are grouped into three classes: \emph{slow} (), \emph{fast} () and \emph{ternary} (). For each category, we derive a \emph{linear ordering} of the set of natural numbers, that captures \emph{forcing} between the \emph{patterns}. We also show that each of these orderings is \emph{stable} under perturbations.

Paper Structure

This paper contains 13 sections, 27 theorems, 6 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Monotonicity
  4. P-adjusted
  5. Loops
  6. Properties of regular patterns
  7. Characterization of Triod-twist patterns
  8. Characterization of regular exact patterns
  9. Properties of Regular exact patterns
  10. Forcing among regular exact patterns
  11. Forcing among exact slow patterns
  12. Forcing among exact fast patterns
  13. Forcing among exact ternary patterns

Key Result

Theorem 1.1

Let $f:[0,1] \to [0,1]$ be a continuous map. If $m,n \in \mathbb{N}$ with $m \succ\mkern-14mu_s\; n$ and $m \in \mathrm{Per}(f)$, then $n \in \mathrm{Per}(f)$. Consequently, there exists some $k \in \mathbb{N} \cup \{2^\infty\}$ such that $\mathrm{Per}(f) = Sh(k)$. Conversely, for every $k \in \math

Figures (7)

  • Figure 1: Schematic representation of modified rotation pairs (mrp) on the real line with attached prongs
  • Figure 2: Bifurcation diagram illustrating the change in color of points with varying rotation number$\rho$.
  • Figure 3: Formation of point loop of rotation pair$(k+1,n+3)$ from the point loop of rotation pair$(k,n)$ in the case, $\frac{k}{n}< \frac{1}{3}$ (See Theorem \ref{['k:n:green:forces:k+1:n+3']})
  • Figure 4: The unimodal slow twist pattern$\Gamma_0^{\frac{2}{9}}$
  • Figure 5: The unimodal fast twist pattern$\Delta_0^{\frac{2}{5}}$
  • ...and 2 more figures

Theorems & Definitions (49)

  • Theorem 1.1: shatr
  • Definition 1.2: mz89zie95
  • Theorem 1.3: alm98BMR
  • Definition 1.4
  • Theorem 1.5: BM2
  • Definition 1.6
  • Theorem 1.7: BB5
  • Theorem 2.1: alm98
  • Theorem 2.2: BMR
  • Theorem 2.3: alm98zie95
  • ...and 39 more