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Dynamical displacements, persistence and semiconjugacies

Philip Boyland

TL;DR

This work develops a unified framework to study asymptotic dynamics on manifolds by extending rotation-set ideas beyond identity isotopies, using both Abelian (via the universal Abelian cover) and non-Abelian (via Nielsen theory) perspectives. It introduces three equivalent constructions yielding semiconjugacies to toral endomorphisms $\Phi_A$ when $f_*$ acts by a matrix $A$, provided $A$ is expanding on homology; these include Bowen–Franks data, global shadowing, and asymptotic displacement limits. A central achievement is the definition and analysis of the direct-limit Bowen–Franks group ${\mathrm{BF}}_{\infty}(A)$, its embedding into the torus, and its connection to the periodic points of $\Phi_A$, which in turn governs the asymptotic dynamics of $f$. The framework extends to non-Abelian Nielsen theory via Handel’s results, yielding semiconjugacies and persistence statements that tie homotopy invariance to the presence of dynamical features across maps in the same homotopy class. Collectively, these methods illuminate how large-scale expansion and shadowing produce robust invariant structures and persistent dynamical patterns across homotopies, with applications to pseudo-Anosov maps and related systems.

Abstract

This survey gives a unified treatment of topics from Abelian and non-Abelian Nielsen Theory integrated with the semiconjugacy theorems of Franks and Handel. The main focus is to develop an analog of the rotation set that is valid when the dynamics are not isotopic to the identity and to connect this theory to the dynamical persistence under homotopy/isotopy intrinsic in the theorems of Franks and Handel. For this dynamical persistence, expansion/hyperbolicity at some scale is essential.

Dynamical displacements, persistence and semiconjugacies

TL;DR

This work develops a unified framework to study asymptotic dynamics on manifolds by extending rotation-set ideas beyond identity isotopies, using both Abelian (via the universal Abelian cover) and non-Abelian (via Nielsen theory) perspectives. It introduces three equivalent constructions yielding semiconjugacies to toral endomorphisms when acts by a matrix , provided is expanding on homology; these include Bowen–Franks data, global shadowing, and asymptotic displacement limits. A central achievement is the definition and analysis of the direct-limit Bowen–Franks group , its embedding into the torus, and its connection to the periodic points of , which in turn governs the asymptotic dynamics of . The framework extends to non-Abelian Nielsen theory via Handel’s results, yielding semiconjugacies and persistence statements that tie homotopy invariance to the presence of dynamical features across maps in the same homotopy class. Collectively, these methods illuminate how large-scale expansion and shadowing produce robust invariant structures and persistent dynamical patterns across homotopies, with applications to pseudo-Anosov maps and related systems.

Abstract

This survey gives a unified treatment of topics from Abelian and non-Abelian Nielsen Theory integrated with the semiconjugacy theorems of Franks and Handel. The main focus is to develop an analog of the rotation set that is valid when the dynamics are not isotopic to the identity and to connect this theory to the dynamical persistence under homotopy/isotopy intrinsic in the theorems of Franks and Handel. For this dynamical persistence, expansion/hyperbolicity at some scale is essential.
Paper Structure (69 sections, 22 theorems, 72 equations, 3 figures)

This paper contains 69 sections, 22 theorems, 72 equations, 3 figures.

Table of Contents

  1. Introduction
  2. The universal Abelian covering space
  3. The covering space
  4. Mapping $\tilde{M}_a$ into ${\mathbb R}^b$
  5. formula for $\tilde{f}$
  6. context
  7. The translation and the case $f_*=I$
  8. the translation
  9. the case $f_* = I$
  10. example
  11. context
  12. Fixed and periodic points
  13. Displacements of fixed points and the Bowen-Franks group
  14. example 2
  15. Comparing fixed points
  16. ...and 54 more sections

Key Result

Lemma 4.1

For $x, x'\in \mathop{\mathrm{Fix}}\nolimits(f)$ the following are equivalent.

Figures (3)

  • Figure 1: Left: The wedge of two circles and its universal Abelian cover. Right: A fundamental domain and its image under the map $\tilde{\phi}_1$.
  • Figure 2: The rotation set of the map $\phi_1$.
  • Figure 3: Left: Approximation to $\tilde{\beta}({\tilde{X}})$ for the map $\phi_2$. Right: Approximation to $\tilde{\beta}({\tilde{X}})$ for the map $\phi_3$ of Example \ref{['Aexpand']} below.

Theorems & Definitions (45)

  • Lemma 4.1
  • Remark 5.1
  • Remark 5.2
  • Theorem 5.3
  • Definition 6.1
  • Lemma 6.2
  • Definition 7.1
  • Lemma 7.2
  • proof
  • Lemma 7.3
  • ...and 35 more