Hölder classifications of finite-dimensional linear flows

TL;DR

The paper provides a complete classification of finite-dimensional linear flows up to several regularity-based notions of equivalence by translating orbit-wise behavior into linear-algebraic data. It introduces and exploits a canonical invariant decomposition $X_S^{\Phi} \oplus X_C^{\Phi} \oplus X_U^{\Phi}$ and Lyapunov similarity to connect time reparametrizations with generator structure, yielding four unified classifications (topological, Hölder, Lipschitz, smooth) for linear flows. Central results (Theorems thma and thmb, and the all-Hölder thm) characterize some-Hölder and all-Hölder equivalences in terms of stable/unstable dimensions, central and hyperbolic block similarities, and Lyapunov exponents (via the Lyapunov cross-ratio $\rho$). The work unifies and extends prior results, clarifies the roles of linearity and finite dimensionality, and extends naturally to complex spaces via realification. Together with companion work, it provides elementary, self-contained proofs and a coherent framework for understanding orbit structure under various regularity constraints.

Abstract

Two flows on a finite-dimensional normed space $X$ are equivalent if some homeomorphism $h$ of $X$ preserves all orbits, i.e., $h$ maps each orbit onto an orbit. Under the assumption that $h$, $h^{-1}$ both are $β$-Hölder continuous near the origin for some (or all) $0<β< 1$, a complete classification with respect to some-Hölder (or all-Hölder) equivalence is established for linear flows on $X$, in terms of basic linear algebra properties of their generators. Consistently utilizing equivalence instead of the more restrictive conjugacy, the classification theorems extend and unify known results. Though entirely elementary, the analysis is somewhat intricate and highlights, more clearly than does the existing literature, the fundamental roles played by linearity and the finite-dimensionality of $X$.

Figures (6)

• Figure 1: No two of the four classifications of all linear flows on $X=\mathbb{R}^2$ coincide.
• Figure 2: Displaying all possible phase portraits (without orientation) of a linear flow $\Phi$ on $X=\mathbb{R}^2$, up to Hölder equivalence (Theorem \ref{['thmb']}). In the bottom half, the two left-most flows are (topologically) equivalent, and so are the two right-most flows (Theorem \ref{['thma']}); see also the lower half of Figure \ref{['fig0a']}.
• Figure 3: Relating eighteen natural equivalences between flows $\varphi$, $\psi$ on $X=\mathbb{R}^d$, $d\in \mathbb{N}$. All equivalences are genuinely different in that no conceivable implication not shown in the diagram is valid in general.
• Figure 4: As a consequence of Theorems \ref{['thma']} and \ref{['thmb']}, as well as Propositions \ref{['thmc']} and \ref{['thmd']}, all equivalences between linear flows $\Phi$, $\Psi$ on $X=\mathbb{R}^d$ coalesce into no more than four different forms.
• Figure 5: Proving Lemma \ref{['lemH2']}(i) by estimating $|P_{\sf S} \Phi_{r} y_r| = |P_{\sf S} h^{-1} \bigl(\Psi_{T_r} h(y_r) \bigr) |$ in two different ways which lead to (\ref{['eqH11']}) and (\ref{['eqH10A']}), respectively.

Theorems & Definitions (49)

• Theorem 1.1
• Theorem 1.2
• Proposition 1.3
• Proposition 1.4
• Proposition 2.1
• Proposition 3.1
• Proposition 3.2
• Remark 3.3
• Lemma 4.1
• proof