Lyapunov exponents for uniformly hyperbolic random matrix products

Abstract: We consider a finite family of invertible $2 \times 2$ real matrices and a transitive Markov shift on the index set. Let $λ$ be the top Lyapunov exponent for random matrix products driven by the Markov shift. We prove that, if the matrices are projectively uniformly hyperbolic with respect to the Markov shift, then $λ$ admits an explicit representation in terms of an infinite matrix. This rapidly convergent representation yields a polynomial-time algorithm for approximating $λ$: only $O\big( (\log(1/\varepsilon))³ \big)$ arithmetic operations are needed to achieve error $\varepsilon$. Furthermore, $λ$ depends real analytically on the matrix entries and the transition probabilities near a projectively uniformly hyperbolic system, and each Taylor coefficient can be approximated in polynomial time.

• The paper presents an infinite-matrix representation for the top Lyapunov exponent, enabling a polynomial-time, ε-accurate computation for 2×2 hyperbolic matrices.
• It leverages a branch-state extension of the Markov process to ensure uniform contraction and demonstrates real-analytic dependence on matrix entries and transition probabilities.
• The method offers certified high-precision estimates and sensitivity analysis with potential applications in fields such as statistical physics and chaos theory.

Explicit Computability of Lyapunov Exponents in Uniformly Hyperbolic Random Matrix Products

Introduction and Context

The paper "Lyapunov exponents for uniformly hyperbolic random matrix products" (2604.12244) addresses the long-standing problem of the explicit and effective computation of the top Lyapunov exponent for random products of matrices driven by a Markov process. While the existence and basic properties of Lyapunov exponents in random dynamical systems are classical, their explicit numerical calculation, even for $2 \times 2$ matrix families, remains notoriously non-trivial due to the intrinsic subadditivity and non-commutativity of the product dynamics.

Building on the geometric theory of projectively uniformly hyperbolic cocycles, the author provides a new infinite-matrix representation for the Lyapunov exponent, specialized to finite families of invertible $2 \times 2$ real matrices under a Markov shift with an associated Markov measure. This representation yields, for the first time in this setting, a polynomial-time algorithm for approximating the Lyapunov exponent up to arbitrary precision in the projectively uniformly hyperbolic regime, and demonstrates real-analytic dependence of the exponent on both the matrix entries and transition probabilities.

Main Results

Infinite-Matrix Representation and Algorithmic Complexity

Let $\{A_i\}_{i \in X}$ be a finite collection of invertible $2 \times 2$ real matrices, and $(\Sigma, P)$ a transitive Markov shift with probability measure $\mu$. The top Lyapunov exponent $\lambda$ is given by:

$$\lambda = \lim_{n \to \infty} \frac{1}{n} \log \| A_{x_{n-1}} \cdots A_{x_0}\|$$

for $\mu$-almost all $(x_n) \in \Sigma$.

Under the assumption that the family $2 \times 2$0 is projectively uniformly hyperbolic (i.e., admits a dominated splitting or, equivalently, a multicone structure in projective dynamics), the author establishes an explicit, rapidly convergent representation of $2 \times 2$1 as an infinite sum involving powers of an explicitly constructed infinite matrix operator $2 \times 2$2:

$2 \times 2$3

Here, each term involves iterates of $2 \times 2$4, acting on a computable seed vector $2 \times 2$5, projected onto a finite-dimensional space with explicit weights dictated by the stationary distribution of the Markov chain.

The combinatorics of the Markov shift and multicone structure are handled by a branch-state extension, which lifts the system to a Markov chain recording both the current branch and multicone location, guaranteeing positivity and uniform contraction in appropriate coordinates.

This representation leads to an algorithm for $2 \times 2$6-accurate computation of $2 \times 2$7, requiring $2 \times 2$8 arithmetic operations. The analysis shows the cubic logarithmic dependence emerges from the convergence properties of the Neumann series associated with the contraction properties of the system and the structure of the matrix truncation.

Analytic Dependence and Taylor Coefficients

A second main result is the real-analytic dependence of the Lyapunov exponent on both the matrix entries and the Markov transition probabilities, in a neighborhood of a projectively uniformly hyperbolic point. The explicit series representation admits termwise differentiation, so the Taylor expansion of $2 \times 2$9 with respect to real-analytic parameters may be computed with the same polynomial efficiency as the exponent itself.

For any fixed derivative order $\{A_i\}_{i \in X}$0, the $\{A_i\}_{i \in X}$1-th derivative of $\{A_i\}_{i \in X}$2 with respect to parameters can be approximated to error $\{A_i\}_{i \in X}$3 with $\{A_i\}_{i \in X}$4 arithmetic operations, with all terms in the expansion accessible and certifiably accurate via interval arithmetic. The paper provides explicit error controls and procedures for the calculation of all higher derivatives.

Examples and Numerical Precision

Detailed computations demonstrate the method's practicality. In an example involving a family generating a non-projectively uniformly hyperbolic system, the theory of Markov harmonic measure (Mairesse) allows reduction to an applicable setting, and explicit formulas produce numerical values for Lyapunov exponents with high-precision certification. Analytic families of matrices and transition probabilities exhibit the ability of the method to produce certified high-order Taylor expansions of $\{A_i\}_{i \in X}$5 in parameter space.

Technical Ingredients

Multicone and Markov Extension

A key device is the multicone criterion, which ensures uniform contraction and positivity in specially chosen projective charts. The extension of the Markov process to a "branch-state" Markov chain remembers the sector in the multicone stratification, allowing the problem to be recast in terms of products of strictly positive matrices—a necessity for the transfer operator approach.

Kernel Expansion and Infinite Matrix Formulation

The core analytic tool is a kernel expansion technique for decomposing the nonlinear functional (variational) expression for $\{A_i\}_{i \in X}$6 into an infinite matrix power series. Each matrix entry can be represented explicitly in terms of the Möbius actions associated with the semigroup generated by the matrices, including their transposes and derivatives with respect to parameters.

Rapid convergence is guaranteed by the explicit contraction rates (dictated by the invariant multicones), and careful analysis provides precise error bounds for truncations.

Polynomial-Time Evaluation

By combining contraction inequalities, operator norm estimates, and a careful analysis of the truncated finite-dimensional approximants to the infinite matrix operator, the total computational effort is shown to grow cubically in the logarithm of the requested precision. The representation is robust, improving on prior approaches (e.g., for positive matrices) that only guaranteed subexponential or worse complexity.

Analyticity and Effective Bounds for Derivatives

The holomorphic extension of relevant quantities (matrix entries, transition probabilities, Möbius actions) is established on a compact neighborhood in parameter space. The infinite-matrix expansion inherits holomorphicity, and termwise differentiation is justified. The Cauchy integral formula provides explicit tail error control for each derivative. The method yields, for each order, explicit and stringent error estimates.

Implications and Future Directions

This work resolves, within the class of projectively uniformly hyperbolic $\{A_i\}_{i \in X}$7 random matrix cocycles (Markov-driven), the algorithmic challenge of effective and certified computation of the top Lyapunov exponent and all its derivatives. The framework provides a conceptual and technical synthesis of transfer operator methods, explicit projective geometry, and modern algorithmic analysis.

Practical Implications:

• Enables robust, high-precision, and efficient computation of Lyapunov exponents in statistical physics, chaos theory, and random dynamical systems wherever projective uniform hyperbolicity can be established.
• Taylor expansions allow sensitivity analysis, parameter studies, and probabilistic stability analysis in numerically stiff regimes.

Theoretical Implications:

• Provides concrete support for the principle that projective uniform hyperbolicity is not merely a qualitative property but admits powerful quantitative and algorithmic consequences.
• Interplay with Markovian dependencies extends the reach of the transfer operator approach, further integrating random products and symbolic dynamics.

Prospects for Future Work:

• Extension to higher-dimensional matrix cocycles and to systems with non-uniform or weaker forms of hyperbolicity.
• Investigation of explicit algorithms for constructing multicones in arbitrary systems, and further automation of the branch-state extension.
• Application to random walks on hyperbolic groups, surface group actions, and the thermodynamic formalism of Markov-driven systems.
• Potential generalizations to non-stationary and non-irreducible Markov processes, or to systems with time-dependent cocycles.

Conclusion

The paper presents a significant advance in the computation and structural understanding of Lyapunov exponents for random matrix products under Markovian driving, when projective uniform hyperbolicity is present. Through an explicit infinite-matrix formulation, the author demonstrates that both the exponent and all of its derivatives can be computed to arbitrary precision in polynomial time, with all constants and convergence rates made explicit. This resolves major computational and conceptual challenges in the field and provides a foundation for further algorithmic and theoretical developments in the quantitative theory of random dynamical systems.