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Generalized dichotomies via time rescaling

Davor Dragicevic, Cesar M. Silva

TL;DR

The paper extends dichotomy theory for nonautonomous linear difference equations to general discrete growth rates by introducing a time-rescaling framework. It proves that a $\mu$-dichotomy with respect to a norm sequence $\{\|\cdot\|_k\}_k$ is equivalent to an ordinary dichotomy for a time-rescaled system and an exponential (or uniform) dichotomy for a discretized operator family $Q^{\mu,\eta}$, under mild regularity of $\mu$. This approach yields a robust spectral theory, showing $\Sigma_{\mu D, \mathbb{A}}=\Sigma_{ED, \mathbb{Q}}$ and transferring the classic Sacker-Sell structure to the generalized setting; it also enables comprehensive nonlinear linearization results, including topological and $C^\ell$ Sternberg-type theorems and $C^1$ linearization under spectral-gap/band conditions. The results unify and extend polynomial, exponential, and nonuniform dichotomies, provide one-sided (half-line) linearization, and offer practical tools for analyzing nonautonomous dynamics with nonexponential growth rates.

Abstract

For discrete-time nonautonomous linear dynamics and a large class of discrete growth rates $μ$, we show that the notion of $μ$ dichotomy (with respect to a sequence of norms) can be completely characterized in terms of ordinary and exponential dichotomy (with respect to a sequence of norms) by employing a suitable rescaling of time. Previously, such a result was known only in the particular case of polynomial dichotomies. As a nontrivial application of our results, we study the structure of a generalized Sacker-Sell spectrum and obtain a series of nonautonomous topological and smooth linearization results.

Generalized dichotomies via time rescaling

TL;DR

The paper extends dichotomy theory for nonautonomous linear difference equations to general discrete growth rates by introducing a time-rescaling framework. It proves that a -dichotomy with respect to a norm sequence is equivalent to an ordinary dichotomy for a time-rescaled system and an exponential (or uniform) dichotomy for a discretized operator family , under mild regularity of . This approach yields a robust spectral theory, showing and transferring the classic Sacker-Sell structure to the generalized setting; it also enables comprehensive nonlinear linearization results, including topological and Sternberg-type theorems and linearization under spectral-gap/band conditions. The results unify and extend polynomial, exponential, and nonuniform dichotomies, provide one-sided (half-line) linearization, and offer practical tools for analyzing nonautonomous dynamics with nonexponential growth rates.

Abstract

For discrete-time nonautonomous linear dynamics and a large class of discrete growth rates , we show that the notion of dichotomy (with respect to a sequence of norms) can be completely characterized in terms of ordinary and exponential dichotomy (with respect to a sequence of norms) by employing a suitable rescaling of time. Previously, such a result was known only in the particular case of polynomial dichotomies. As a nontrivial application of our results, we study the structure of a generalized Sacker-Sell spectrum and obtain a series of nonautonomous topological and smooth linearization results.
Paper Structure (9 sections, 13 theorems, 170 equations)

This paper contains 9 sections, 13 theorems, 170 equations.

Table of Contents

  1. Introduction
  2. Time rescaling
  3. $\mu$-dichotomy
  4. Dichotomy spectrum
  5. Nonautonomous topological linearization
  6. Nonautonomous smooth linearization: A Sternberg-type theorem
  7. Smooth linearization under spectral gap and band conditions
  8. Appendix
  9. Nonautonomous topological linearization on the half-line

Key Result

Theorem 2.2

Let $\left \{\| \cdot \|_k \right \}_{k\in \mathbb{N}}$ be a sequence of norms on $X$ and $\mu$ a growth rate satisfying for some $\theta\ge 1$. The following assertions are equivalent:

Theorems & Definitions (34)

  • Definition 2.1
  • Definition 2.2
  • Remark 2.1
  • Theorem 2.2
  • proof
  • Remark 2.3
  • Example 2.4
  • Theorem 2.5
  • proof
  • Definition 3.1
  • ...and 24 more