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Inheritance of shadowing for dynamical semigroups

Michael Blank

TL;DR

This work extends the single perturbation shadowing framework to semigroups of endomorphisms and to non-autonomous systems, introducing a gluing construction that yields shadowing results from a single perturbation property. It shows that shadowing inheritance between generators and the generated semigroup is not generally valid, providing explicit counterexamples, and highlights important differences between semigroups and non-autonomous dynamics. The results formalize when shadowing for all generators implies shadowing for the semigroup and when it does not, with precise conditions expressed through rate functions $\varphi$ and summability, and they connect uniform and average shadowing notions via the ${\cal S}(\alpha,\beta)$ framework. The findings have implications for the numerical analysis of chaotic semigroup dynamics and for understanding robustness of shadowing under generator reparametrizations and time-dependent perturbations.

Abstract

We extend the single-perturbation approach (developed in our earlier publications for the case of a single map) to the analysis of the shadowing property for semigroups of endomorphisms. Our approach allows to give a constructive representation for a true trajectory which shadows a given pseudo-trajectory. One of the main motivations is the question of inheritance: does the presence of shadowing for all generators of a semigroup imply shadowing for the semigroup and vice versa. Somewhat surprisingly, the answer to these questions is generally negative. Moreover, the situation with shadowing turns out to be quite different in a semigroup and in a non-autonomous system, despite the fact that the latter can be represented as a single branch of the former.

Inheritance of shadowing for dynamical semigroups

TL;DR

This work extends the single perturbation shadowing framework to semigroups of endomorphisms and to non-autonomous systems, introducing a gluing construction that yields shadowing results from a single perturbation property. It shows that shadowing inheritance between generators and the generated semigroup is not generally valid, providing explicit counterexamples, and highlights important differences between semigroups and non-autonomous dynamics. The results formalize when shadowing for all generators implies shadowing for the semigroup and when it does not, with precise conditions expressed through rate functions and summability, and they connect uniform and average shadowing notions via the framework. The findings have implications for the numerical analysis of chaotic semigroup dynamics and for understanding robustness of shadowing under generator reparametrizations and time-dependent perturbations.

Abstract

We extend the single-perturbation approach (developed in our earlier publications for the case of a single map) to the analysis of the shadowing property for semigroups of endomorphisms. Our approach allows to give a constructive representation for a true trajectory which shadows a given pseudo-trajectory. One of the main motivations is the question of inheritance: does the presence of shadowing for all generators of a semigroup imply shadowing for the semigroup and vice versa. Somewhat surprisingly, the answer to these questions is generally negative. Moreover, the situation with shadowing turns out to be quite different in a semigroup and in a non-autonomous system, despite the fact that the latter can be represented as a single branch of the former.
Paper Structure (10 sections, 9 theorems, 39 equations, 2 figures)

This paper contains 10 sections, 9 theorems, 39 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Setting and main result
  3. Proof of Theorem \ref{['t:generators']}
  4. Proof of Theorem \ref{['t:main']}
  5. Shadowing for non-autonomous systems
  6. Inheritance of shadowing properties
  7. Conjugation property and shadowability
  8. The simplest examples of shadowable and non-shadowable maps
  9. Inheritance of shadowing from generators to semigroups and backwards
  10. Acknowledgment

Key Result

Theorem 2.1

Let $G$ be a semigroup of maps from a complete metric space $(X,\rho)$ into itself. Then if for some choice of generators $G\in {\cal S}(\alpha,\beta)$ with $\alpha\in\{U,A,S\},~ \beta\in\{U,A,L\}$, then the same property holds true for any other choice of generators.

Figures (2)

  • Figure 1: Order of the parallel gluing.
  • Figure 2: Contributions to the upper bound of the gluing error.

Theorems & Definitions (27)

  • Definition 2.1
  • Definition 2.2
  • Remark 2.1
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Definition 2.7
  • Definition 2.8
  • Remark 2.2
  • ...and 17 more