Discontinuity of Lyapunov Exponent

• Discontinuity of Lyapunov exponent is a phenomenon where minute parameter changes cause sudden shifts in the rate of growth, impacting stability and spectral properties.
• Mechanisms such as localized resonance, hyperbolicity destruction, and spectral interlacing illustrate how small perturbations can result in significant dynamical and bifurcation effects.
• The interplay of arithmetic properties, regularity classes, and topological conditions critically governs these discontinuities, offering insights into phase transitions and system robustness.

The discontinuity of the Lyapunov exponent is a phenomenon in dynamical systems, random matrix products, and spectral theory in which small perturbations of system parameters—such as cocycle maps, potentials, or probability distributions—lead to abrupt jumps in the value of the Lyapunov exponent. This behavior stands in sharp contrast to regimes where the Lyapunov exponent depends continuously, or even smoothly, on underlying parameters. The existence, nature, and structure of such discontinuities depend on precise regularity, topology, arithmetic properties, and moment conditions, with significant implications for dynamical stability, phase transitions, and spectral phenomena.

1. Formal Definitions and Types of Discontinuity

Let $(X, f)$ be a dynamical system (such as a compact manifold with a smooth or continuous map) and $A: X \to G$ a (linear or non-linear) cocycle taking values in a Lie group (typically $SL(2,\mathbb{R})$ or $GL(d,\mathbb{R})$). Given a probability measure $\mu$ invariant and ergodic under $f$, the top Lyapunov exponent is defined as

$$\lambda_+(A, \mu) = \lim_{n \to \infty} \frac{1}{n} \int_X \log \|A^n(x)\|\, d\mu(x)$$

where $A^n(x)$ is the product cocycle along the orbit. Discontinuity occurs if there exists a sequence $A_k \to A$ in a topology (e.g., $C^0$, $C^\alpha$, $C^l$, analytic, Gevrey, Wasserstein, or weak-$*$ for measures) with

$$\lim_{k \to \infty} \lambda_+(A_k, \mu) \neq \lambda_+(A, \mu).$$

Analogous definitions apply for Lyapunov exponents $L(E)$ as functions of energy $E$ for operators (e.g., Schrödinger operators), or as functions of the underlying measure or potential.

Discontinuity is often contextualized further:

• Discontinuity in parameter: The exponent jumps under small changes to a system parameter (potential, map, etc.).
• Discontinuity in topology: The exponent is not continuous in the normed topology (e.g., $C^l$ vs. analytic/Gevrey) or measure topology (Wasserstein/weak-$*$).
• Discontinuity as a function: The exponent $L(E)$ as function of energy can be discontinuous on a set of positive measure, as in limit-periodic Schrödinger operators.

2. Topological and Regularity Conditions Governing Discontinuity

The existence of discontinuity in Lyapunov exponents is highly sensitive to the topology and regularity class:

Setting	Continuity Regime	Discontinuity Regime
Analytic/monotonic cocycles (Diophantine)	Continuous (analytic $\Rightarrow$ LE continuous)	None known
Gevrey $G^s$ ($s<2$), strong Diophantine	Continuous	Not continuous for $s>2$
$C^l$, $l<\infty$ ($SL(2,\mathbb{R})$ cocycles)	Generic continuity, but discontinuities exist (Wang et al., 2012)	Constructed everywhere, even $C^\infty$ (Wang et al., 2012)
Hölder cocycles (fiber bunched)	Continuous [Backes–Brown–Butler]	Discontinuous near boundary (Butler, 2015, Mamani et al., 29 Jul 2024)
Hölder cocycles (non-fiber bunched)	Not continuous (discontinuity points dense)	Discontinuities explicit
Probability measures: compact support	Continuous or semi-continuous [Le Page]	Pathologies unlikely
Probability measures: non-compact support	Not continuous, only upper/lower semi-continuity in Wasserstein (Sánchez et al., 2018, Duarte et al., 8 Sep 2025)	Discontinuities constructed

Continuity in the analytic or sub-Gevrey-2 category is established for a broad class of quasi-periodic cocycles and Schrödinger operators. The Gevrey $G^2$ class appears as a sharp threshold: the Lyapunov exponent is always continuous for $G^s$, $s<2$, but examples in $G^s$ for $s>2$ (and, for more Liouvillean frequencies, even in $C^\infty$) exhibit discontinuity (Liang et al., 28 Oct 2025, Ge et al., 2021). For $C^l$ (finite or infinite) smooth cocycles, explicit construction of discontinuity points are given, even with frequency of bounded-type (Wang et al., 2012).

For random matrix products, Lyapunov exponents regarded as functions of the underlying probability measure are generally only upper (or lower) semi-continuous in strong topologies (Wasserstein), but are not continuous, and can be highly discontinuous for weak-* convergence or without moment bounds (Sánchez et al., 2018, Duarte et al., 8 Sep 2025).

3. Mechanisms and Constructions Yielding Discontinuity

Several different mechanisms are employed in the construction of discontinuities:

• Localized Resonance: In quasi-periodic cocycles, constructing smooth cocycles that differ from a hyperbolic reference cocycle only on very small intervals leads to drastically lowered Lyapunov exponent due to critical alignments or near-reducibility on these intervals (Wang et al., 2012, Liang et al., 28 Oct 2025, Ge et al., 2021). The flatness or smoothness of the perturbation (bump functions) directly governs which regularity class allows discontinuity.
• Destruction of Hyperbolicity: In locally constant $SL(2,\mathbb{R})$ cocycles over the Bernoulli shift, small perturbations supported on rare cylinders can kill the exponential growth of the cocycle, driving the exponent abruptly to zero, even though the cocycle remains arbitrarily close in the Hölder norm (Butler, 2015, Mamani et al., 29 Jul 2024).
• Spectral Interlacing: For limit-periodic Schrödinger operators, one can arrange for the spectrum to be densely covered by bands with vanishingly small Lyapunov exponent, but with positive exponents persisting on neighboring spectral sets, yielding (in the limit) a Lyapunov exponent that is discontinuous on a set of positive measure (Gan et al., 2010).
• Random Matrix Products with Heavy Tails: For probability measures on matrices with heavy (non-integrable or insufficient) tails, rare but extremely large expansions dominate, allowing for constructed sequences of measures converging (in Wasserstein or weak-*) where the Lyapunov exponent jumps—sometimes from zero to positive, or vice versa (Sánchez et al., 2018, Duarte et al., 8 Sep 2025).
• Bifurcation in Holomorphic Maps: For complex polynomials or exponential maps with a single critical point, the Lyapunov exponent at the critical value is discontinuous at the parameter values corresponding to the appearance or disappearance of attracting cycles (Levin et al., 2013).

4. Discontinuity as a Probe of Dynamical and Spectral Transitions

Discontinuities in the Lyapunov exponent often mark underlying structural transitions:

• Thermodynamic Phase Transitions in Black Holes: In charged regular AdS black holes or Schwarzschild AdS black holes with a cloud of strings, the Lyapunov exponent for unstable circular geodesics rapidly jumps across the coexistence line between small and large black hole phases—mirroring the discontinuity of order parameters in standard thermodynamic phase transitions. Near the critical point, the difference exhibits universal mean-field scaling, further reinforcing its role as a dynamical order parameter for black hole phase structure (Xie et al., 27 Oct 2025, Kumar et al., 4 Aug 2025).
• Entropy Spectrum and Multifractality: In hyperbolic systems, the multifractal entropy spectrum of Lyapunov exponents can itself be discontinuous at special points on the boundary of the allowable exponent set, even when it is analytic and concave in the interior (Javornik et al., 2019). This reflects the subtlety of multifractal analysis and the influence of geometric properties (polyhedrality) of the Lyapunov exponent set.
• Regularization by Noise: For random dynamical systems on compact manifolds, the deterministic system can support a Lyapunov irregular set of positive measure (points where exponents do not exist), but any physical noise collapses this irregular set to measure zero and regularizes the infinitely many possible local exponents into a finite collection, with exponents existing almost everywhere (Nakamura et al., 2021).

5. Interplay of Arithmetic, Regularity, and Topology

A critical aspect governing discontinuity is the interaction between arithmetic properties (e.g., Diophantine, Brjuno, Liouvillean frequency types), regularity (analytic, Gevrey, $C^l$), and topology (norms, measure). For instance:

• For strong Diophantine frequencies, the transition from continuity to discontinuity as one crosses from $G^s$ for $s<2$ to $s>2$ is sharp (Liang et al., 28 Oct 2025, Ge et al., 2021).
• For weaker Diophantine or more Liouvillean frequencies, discontinuity emerges in less smooth spaces: for Brjuno frequencies in $C^\infty$, or finite-Liouvillean in finite $C^l$ (Liang et al., 28 Oct 2025).
• The precise modulus of continuity for the Lyapunov exponent as a function of measure is strictly controlled by the integrability (moment) profile; exponential moment is necessary for Hölder continuity, with sub-exponential or polynomial moments allowing only correspondingly weaker regularity, and complete breakdown in the absence of such bounds (Duarte et al., 8 Sep 2025).

6. Consequences, Open Problems, and Broader Impact

Discontinuity of Lyapunov exponents constrains the stability, rigidity, and bifurcation structure of dynamical systems and operators:

• Sharpness of Structural Stability: In the context of cocycles, boundary regimes (e.g., the fiber-bunching threshold in Hölder cocycles) are proved to be optimal for the persistence of Lyapunov exponent continuity (Butler, 2015, Mamani et al., 29 Jul 2024).
• Dynamical Diagnosis of Criticality: The abrupt change and universal scaling of Lyapunov exponents near thermodynamic critical points in black holes tie dynamical instability directly to phase structure, emphasizing the Lyapunov exponent as a physical observable (Xie et al., 27 Oct 2025, Kumar et al., 4 Aug 2025).
• Spectral and Transport Properties: For limit-periodic and more general non-periodic systems, the presence of discontinuities challenges the applicability of Kotani theory, complicating spectral decompositions and questions of absolute continuity (Gan et al., 2010).

Several open problems persist:

• Is the Gevrey $G^2$ transition for continuity/discontinuity of Lyapunov exponents universal for all frequency types?
• Can a complete dictionary be given relating measure moment profile to optimal modulus of Lyapunov exponent continuity?
• How do similar discontinuity phenomena interact with regularity of the integrated density of states or spectral measure in quasi-periodic and random operators?

7. Notable Examples and Summary Table

Context	Discontinuity Manifestation	References
Quasiperiodic cocycles ($C^l$, $G^s$)	Existence of cocycles where LE is not continuous	(Wang et al., 2012, Liang et al., 28 Oct 2025, Ge et al., 2021)
Locally constant cocycles (Hölder)	Discontinuity near but outside fiber-bunching	(Butler, 2015, Mamani et al., 29 Jul 2024)
Limit-periodic Schrödinger operators	LE discontinuous on sets of positive measure in $E$	(Gan et al., 2010)
Random matrices (non-compact support)	Only upper/lower semi-continuity in $W_1$, not continuity; explicit pathologies	(Sánchez et al., 2018, Duarte et al., 8 Sep 2025)
Holomorphic dynamics	Jump in LE at catalysis of attracting cycles	(Levin et al., 2013)
Black hole dynamics	LE acts as dynamical order parameter, jumps at transition	(Xie et al., 27 Oct 2025, Kumar et al., 4 Aug 2025)

The discontinuity of the Lyapunov exponent serves both as a diagnostic and a limitation: it reveals phase transitions, subtle dynamical/spectral bifurcations, and the influence of arithmetic and regularity, while simultaneously restricting the domain of validity for perturbative and qualitative stability results. Its rigorous mathematical understanding continues to influence diverse areas—ergodic theory, smooth dynamics, random and quasi-periodic operators, and gravitational physics.