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Dichotomies uniform on subspaces and formulas for dichotomy spectra

Adam Czornik, Konrad Kitzing, Stefan Siegmund

Abstract

In this note we introduce a notion of dichotomy which generalizes the classical concept of exponential dichotomy and the recent notion of Bohl dichotomy. A key attribute is the discussion of the sets of subspaces of the state space on which the dichotomy estimates are uniform. Two main results are a dichotomy spectral theorem based on our notion of dichotomy which is uniform on subspaces and a formula for the dichotomy spectral intervals which is new for the Bohl dichotomy spectrum as well as for the classical exponential dichotomy spectrum.

Dichotomies uniform on subspaces and formulas for dichotomy spectra

Abstract

In this note we introduce a notion of dichotomy which generalizes the classical concept of exponential dichotomy and the recent notion of Bohl dichotomy. A key attribute is the discussion of the sets of subspaces of the state space on which the dichotomy estimates are uniform. Two main results are a dichotomy spectral theorem based on our notion of dichotomy which is uniform on subspaces and a formula for the dichotomy spectral intervals which is new for the Bohl dichotomy spectrum as well as for the classical exponential dichotomy spectrum.

Paper Structure

This paper contains 5 sections, 16 theorems, 99 equations.

Table of Contents

  1. Introduction
  2. Dichotomies
  3. Bohl exponents and dichotomy estimates
  4. The spectral theorem
  5. Maximal subspaces of uniformity

Key Result

Theorem 6

If system 1 has a dichotomy on $(L_1,L_2)$ uniformly on the subspaces in $(\mathcal{U}_1,\mathcal{U}_2)$, then it also has a dichotomy on $(L_1,L_2)$ uniformly on the subspaces in the refinement $(\mathcal{V}_1,\mathcal{V}_2)$ with

Theorems & Definitions (53)

  • Definition 1: Exponential and Bohl dichotomy
  • Remark 2: History of dichotomy notion
  • Remark 3: Uniformity subspaces of Bohl and exponential dichotomy
  • Definition 4: Dichotomy uniform on subspaces of a splitting
  • Remark 5: Examples of dichotomies uniform on subspaces of a splitting
  • Theorem 6: Refining uniformity subspaces of dichotomy
  • proof
  • Remark 7: Dichotomy is uniform on one-dimensional subspaces
  • Theorem 8: Exponential and Bohl dichotomy spectrum
  • Definition 9: Dichotomy estimates
  • ...and 43 more