Robustness of Exponential Dichotomies

• Tempered exponential dichotomy is a robust invariant splitting property that characterizes nonuniform hyperbolicity in linear cocycles, evolution families, and random systems in Banach spaces.
• The approach uses admissibility techniques and contraction mappings in weighted function spaces to ensure persistence under small linear perturbations with explicit control over dichotomy constants and projections.
• Quantitative stability estimates reveal how perturbation size influences dichotomy parameters, establishing sharp bounds applicable to discrete, continuous, and stochastic system classes.

A tempered exponential dichotomy is a robust invariant splitting property characterizing nonuniform hyperbolicity for linear cocycles, evolution families, and random dynamical systems in Banach spaces. The robustness property asserts that tempered (or more generally, nonuniform) exponential dichotomies persist under sufficiently small linear perturbations of the underlying operators, with explicit control over the dichotomy constants and projections. This principle has been established with full generality using admissibility techniques, contraction mappings in weighted function spaces, and norm equivalence arguments, encompassing discrete, continuous, stochastic, random, and boundary-value problem settings (Chemnitz et al., 23 Sep 2024, Dragičević et al., 2022, Dragicevic, 14 Dec 2025, Dragicevic et al., 17 Oct 2024, Silva, 2020).

1. Definition and Characterization of Exponential Dichotomy Robustness

Let $(\Omega,\mathcal F,\mathbb P,\sigma)$ be a probability space with an ergodic, aperiodic, measure-preserving transformation, and $X$ a separable Banach space. A linear cocycle $$\mathcal A:\Omega\times\mathbb N\rightarrow\mathcal L(X)$$ admits a tempered exponential dichotomy if there exist a full-measure, $\sigma$-invariant set $\Omega'$, a measurable family of projections $\Pi^s(\omega)$, real $\lambda>0$, and tempered $K:\Omega\to(0,\infty)$ such that for all $\omega\in\Omega'$ and $n\ge0$: $$\|\mathcal A(\omega,n)\,\Pi^s(\omega)\|\le K(\omega)e^{-\lambda n}, \quad \|\mathcal A(\omega,-n)\,\Pi^u(\omega)\|\le K(\omega)e^{-\lambda n},$$ with invariance of the splitting and measurability conditions as detailed in (Chemnitz et al., 23 Sep 2024). The robustness property states: any cocycle $\mathcal B$ with generator $B(\omega)$ satisfying

$\|A(\omega)-B(\omega)\|\le c(\omega)\qquad\text{(with %%%%12%%%% tempered and sufficiently small)},$

admits a tempered exponential dichotomy, possibly with modified projections and constants.

2. Admissibility and Fixed-Point Framework

The proof of robustness relies on the admissibility of pairs of weighted function spaces. For $\mathcal A$ with a dichotomy, the associated input–output operator between spaces like $L^\infty(\Omega,X)$ and $L^\infty_K(\Omega,X)$ is invertible (Mather-type admissibility). For the perturbed cocycle, one formulates the inhomogeneous difference equation as a fixed-point problem: $$f(\omega)-B(\sigma^{-1}\omega)f(\sigma^{-1}\omega)=g(\omega).$$ The map $\mathcal T$ defined via (weighted) sums over stable/unstable projections is shown to be a contraction under explicit smallness conditions on the norm of $B-A$ (Chemnitz et al., 23 Sep 2024, Dragičević et al., 2022, Dragicevic, 14 Dec 2025, Dragicevic et al., 17 Oct 2024, Silva, 2020). The contraction-mapping principle ensures existence and uniqueness of a solution, establishing admissibility for the perturbed system, which via converse admissibility theorems implies the existence of a tempered exponential dichotomy.

System Class	Projection Required	Smallness Condition
Discrete cocycles	Measurable, tempered	$\|A_n-B_n\|\le c(n)$
Continuous evolution families	Norm-continuous	$\|A(t)-B(t)\|\leq \delta e^{-\eta t}$
Random dynamical systems	Measurable, random	$$\|\Delta(1,\ell,\omega)\|\leq \varepsilon(\ell,\omega)$$
Stochastic systems (mean-square)	Orthogonal	$\|B(t)\|,\|H(t)\|$ exponentially small

3. Functional and Perturbation Classes

Robustness results extend beyond classical bounded perturbations:

• Arbitrary growth rates (including polynomial, exponential, logarithmic) are covered via generalized admissibility (Dragicevic et al., 17 Oct 2024).
• Evolution families may be noninvertible, and nonuniform weights (e.g., $e^{\varepsilon|t|}$) are allowed, yielding nonuniform exponential dichotomies (Dragicevic, 14 Dec 2025).
• Perturbations may be measured in operator norm, weighted $L^q$ norms, or via convolution bounds: $$\sup_t \int e^{-\alpha|t-s|}\|B(s)\|e^{\varepsilon|s|}ds < \frac{1}{2K}$$ guarantees preservation of the dichotomy (Dragicevic, 14 Dec 2025).

4. Explicit Parameter Dependence, Stability Estimates, and Sharpness

Quantitative upper bounds for perturbed dichotomy constants, rates, and projections are provided. Typical statements include:

• The perturbed exponent $\lambda'$ satisfies $\lambda'-\lambda=O(\delta)$ for perturbation size $\delta$.
• The bound $K'$ inflates as $K' = K/(1-\theta)$, with $\theta=K\delta/(e^\lambda-1)$, and blows up as the perturbation approaches the threshold (Silva, 2020).
• Projections depend Lipschitz or Hölder-continuously on perturbation parameters; e.g.,

$$\|\Pi^{\xi_2}(\omega)-\Pi^{\xi_1}(\omega)\|\leq C(\omega)\|\xi_2-\xi_1\|^\sigma$$

for parametric perturbations of random cocycles (Dragičević et al., 2022).

• Sharpness: the smallness condition cannot generally be weakened. If $K\delta\geq e^\lambda-1$, dichotomy persistence fails; Neumann series or resolvent fails to converge (Silva, 2020, Dragičević et al., 2022).

5. Generalizations, Examples, and Extensions

Robustness applies in broad settings:

• Infinite-dimensional Banach spaces, stochastic systems (mean-square dichotomies), time-dependent and boundary-value problems (Zhu, 2019, Kmit et al., 2013).
• Examples include systems with explicit nonuniform rates, such as cocycles on irrational rotations or hyperbolic PDEs with reflection boundary conditions, for which admissibility and smoothing estimates are verified (Dragičević et al., 2022, Kmit et al., 2013).
• For nonlinear systems, robustness of nonuniform hyperbolicity persists under suitably small $C^1$ nonlinear perturbations, as established through linearization and isolation radius arguments (Caraballo et al., 2020).

6. Connections to Other Nonuniform Dichotomy Theories

Tempered exponential dichotomy is a special case within a hierarchy of generalized nonuniform dichotomies (e.g., $(\mu,\nu)$-dichotomies, polynomial/logarithmic rates). Robustness arguments via admissibility apply uniformly, covering polynomial and logarithmic cases (Backes et al., 6 Jun 2024, Dragicevic et al., 2020, Dragicevic et al., 17 Oct 2024, Silva, 2020).

• Recent advances have removed dependence on Lyapunov norms or compactness, using input-output solvability and fixed-point strategies in weighted function spaces (Dragičević et al., 2022, Backes et al., 6 Jun 2024).
• The dichotomy persists under broader classes of perturbations, including those not bounded in operator norm but integrable in appropriate weights (Dragicevic, 14 Dec 2025).

7. Summary of Principal Results

Robustness of (tempered/nonuniform) exponential dichotomies is a consequence of the invertibility of admissibility operators under small perturbations in weighted Banach spaces. Perturbed systems retain exponential/integral splitting, with explicit stability of dichotomy constants and projections. These results encompass deterministic, stochastic, random, and infinite-dimensional systems, and have eliminated previous technical restrictions such as bounded growth, compactness, or Lyapunov norm construction. The mechanism is always smallness of the perturbation yielding a contraction mapping or Neumann-series invertibility, followed by reassembly of the splitting via admissibility (Chemnitz et al., 23 Sep 2024, Dragičević et al., 2022, Dragicevic, 14 Dec 2025, Dragicevic et al., 17 Oct 2024, Silva, 2020).