Spectral Characterization for Shadowing

• Spectral Characterization for Shadowing is the study of invariant spectral properties that determine the ability of pseudo-orbits to closely follow true trajectories in dynamical systems.
• It establishes a precise spectral criterion, where the right spectrum’s exclusion from the unit circle ensures shadowing and distinguishes it from hyperbolicity and uniform expansivity.
• This framework bridges operator theory and ergodic methods, extending classical stability analysis to infinite-dimensional, nonlinear, and group action dynamical systems.

Spectral characterization for shadowing addresses the precise relationship between shadowing properties in dynamical systems and spectral features of the associated operators, especially in infinite-dimensional, topological, and group action contexts. Shadowing—requiring approximate trajectories ("pseudo-orbits") to stay close to exact orbits—serves as a fundamental bridge between stability, expansivity, and spectral decomposition in dynamical systems. The role of spectral characterization is to identify invariant spectral conditions that guarantee or obstruct shadowing, thus integrating operator theory and ergodic theory in the analysis of linear and nonlinear dynamical behavior.

1. Shadowing in Dynamical Systems: Definitions and Core Notions

Let $T\in L(X)$ be an invertible bounded linear operator on a complex Hilbert space $X$. For $\delta>0$, a bi-infinite sequence $\{x_n\}_{n\in \mathbb{Z}}\subset X$ is a $\delta$–pseudo-trajectory of $T$ if $\|x_{n+1}-T x_n\|\le \delta$ for all $n$. $T$ has the shadowing property if for every $\varepsilon>0$, there exists $\delta>0$ such that every $\delta$–pseudo-trajectory is $\varepsilon$–shadowed by a genuine orbit, meaning there is $x\in X$ with $\|x_n - T^n x\|\le \varepsilon$ for all $n$ (Pituk, 15 Nov 2025). Analogous definitions extend to continuous maps of compact metric spaces and to group actions, with pseudo-orbit and shadowing definitions adapted to the relevant setting (Kawaguchi, 2023, Khan et al., 2018).

Several refinements and generalizations are considered:

• Lipschitz and contractive shadowing: $f$ has $L$-Lipschitz shadowing if every $\delta$-pseudo-orbit is $L\delta$-shadowed for $L>0$, and contractive if $L<1$ (Kawaguchi, 2023).
• L-shadowing: For all $\varepsilon>0$, one requires $\delta>0$ such that every sequence with $d(f(x_k),x_{k+1})\le\delta$ and $d(f(x_k),x_{k+1})\to 0$ as $|k|\to \infty$ can be both $\varepsilon$- and asymptotically-shadowed in the two-sided sense (Artigue et al., 2019).
• Sequential shadowing property (SSP): In group actions, bi-infinite sequences are traced sequentially using group elements, ensuring that shadowing implicates transitive decomposition (Khan et al., 2018).
• Operator-theoretic shadowing: For diffeomorphisms, shadowing is recast as invertibility of a “tangent-orbit” operator on Banach or Fréchet sequence spaces (Dragicevic et al., 2011).

2. Spectral Criteria for Shadowing in Linear Operators

A central result is the complete spectral characterization of shadowing for an invertible bounded linear operator $T$ on a complex Hilbert space:

Theorem: $T$ has the shadowing property if and only if its right spectrum $\sigma_r(T)$ is disjoint from the unit circle $\mathbb{T}=\{z\in\mathbb{C}: |z|=1\}$; that is,

$$T \text{ has the shadowing property } \iff \sigma_r(T)\cap \mathbb{T} = \emptyset.$$

(Pituk, 15 Nov 2025)

The right spectrum $\sigma_r(T)$ consists of those $\lambda$ for which $\lambda I - T$ fails to be right-invertible in $L(X)$. By duality, this is equivalent to $$\lambda\notin \sigma_r(T)\iff\overline{\lambda}\notin\sigma_a(T^*)$$, where $\sigma_a(T^*)$ is the approximate point spectrum of the adjoint.

This criterion distinguishes shadowing from hyperbolicity ($\sigma(T) \cap \mathbb{T} = \emptyset$) and uniform expansivity ($\sigma_a(T) \cap \mathbb{T} = \emptyset$).

Table: Spectral Criteria for Different Properties

Property	Spectral Criterion	Reference
Shadowing (for $T$)	$\sigma_r(T)\cap\mathbb{T}=\emptyset$	(Pituk, 15 Nov 2025)
Uniform expansivity	$\sigma_a(T)\cap\mathbb{T}=\emptyset$	(Pituk, 15 Nov 2025)
Hyperbolicity	$\sigma(T)\cap\mathbb{T}=\emptyset$	(Pituk, 15 Nov 2025)

This framework is functional-analytic, connecting operator invertibility (for certain spectrum) directly to the shadowing property via properties of shift operators and their adjoints.

3. Operator-Theoretic and Banach/Fréchet Space Approaches

In the general (nonlinear) context, spectral characterization of shadowing can be formulated using the invertibility of associated "tangent-orbit" operators. Consider $f:M\to M$ a $C^1$ diffeomorphism, and sequences $v=(v_n)_{n\in\mathbb{Z}}$ in the tangent bundle. The operator $\mathcal{L}v_n = Df(p_{n-1})v_{n-1} - v_n$ acts on spaces such as $\ell^\infty$ (bounded sequences) or weighted Banach/Fréchet spaces of sub-exponential growth (Dragicevic et al., 2011).

• Uniform hyperbolicity: Equivalent to invertibility of $\mathcal{L}$ on $\ell^\infty$ uniformly over the invariant set.
• Nonuniform hyperbolicity: Characterized by invertibility of $\mathcal{L}$ on sub-exponentially weighted spaces $\mathcal{N}$ almost everywhere with respect to an invariant measure.

The generalized shadowing lemma in this framework asserts that if $\mathcal{L}$ is invertible with uniform bounds on an invariant set $A$, then all sufficiently accurate pseudo-orbits remaining near $A$ are shadowed by true orbits, with error scaling to zero as the pseudo-orbit deviation vanishes.

This operator perspective subsumes classical shadowing results (e.g., for Anosov systems) and extends to nonuniformly hyperbolic and even boundary cases (Dragicevic et al., 2011).

4. Shadowing and Spectral Decomposition Theorems

Shadowing properties, when combined with expansivity or suitable generalizations, induce rich spectral (or “Smale–Bowen type”) decompositions:

• Contractive (Lipschitz) shadowing ($L<1$): Ensures the chain-recurrent set $\mathrm{CR}(f)$ of a continuous self-map of a compact metric space splits into finitely many closed chain components. Each component, if chain-transitive, further admits a finite cyclic decomposition into open-closed sets on which $f^m$ is chain-mixing and supports the same $L$-Lipschitz shadowing (Kawaguchi, 2023).
• L-shadowing property: Guarantees a finite decomposition of the chain-recurrent set into chain-recurrent classes, each either expansive (thus “basic set” in the hyperbolic sense) or containing arbitrarily small topological semi-horseshoes carrying positive topological entropy. The classical Smale–Bowen decomposition for hyperbolic homeomorphisms is thus subsumed by L-shadowing (Artigue et al., 2019).
• Sequential shadowing for group actions: The presence of the sequential shadowing property (SSP) suffices for a finite partition of the state space into closed, invariant, transitive components, even in the absence of expansivity (Khan et al., 2018).

5. Connections Among Expansivity, Shadowing, and Uniform Expansivity

A key result in Hilbert space dynamics links the shadowing property of an operator $T$ to uniform expansivity of the adjoint $T^*$. Specifically,

$$T \text{ has the shadowing property} \iff T^* \text{ is uniformly expansive}.$$

This equivalence is effected through the relations of the right spectrum of $T$ and the approximate point spectrum of $T^*$ (Pituk, 15 Nov 2025). The contrast between the spectral locations responsible for hyperbolicity, uniform expansivity, and shadowing is highlighted through explicit examples (e.g., bilateral weighted shifts) exhibiting one property but not others.

6. Illustrative Counterexamples and Example Classes

The distinctions among spectral criteria are exhibited via concrete operator examples:

• Bilateral weighted shifts: Operators can be uniformly expansive without shadowing, or possess shadowing without uniform expansivity, illustrating the sharpness and independence of the spectral criteria for shadowing, uniform expansivity, and hyperbolicity (Pituk, 15 Nov 2025).
• Symbolic and ultrametric dynamics: Full shifts and their analogues on ultrametric spaces serve as examples where contractive shadowing yields trivial spectral decompositions (single mixing basic set), further exhibiting the broad applicability of the contractive/shadowing framework (Kawaguchi, 2023).

7. Broader Implications and Extensions

Spectral characterization of shadowing unifies operator theory and dynamical systems, providing necessary and sufficient conditions for robust approximation of orbits by pseudo-orbits across linear, nonlinear, and group action settings. These results extend the structural theory of Smale, Bowen, and Mather to infinite dimensions, nonuniformly hyperbolic, and nonexpansive regimes, underpinning modern stability theory, ergodic decomposition, and revealing the precise operator-theoretic mechanisms by which topological and metric shadowing properties emerge (Pituk, 15 Nov 2025, Dragicevic et al., 2011, Kawaguchi, 2023, Artigue et al., 2019, Khan et al., 2018).