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arxiv: 2604.12244 · v1 · submitted 2026-04-14 · 🧮 math.DS

Lyapunov exponents for uniformly hyperbolic random matrix products

Nima Alibabaei This is my paper

Pith reviewed 2026-05-10 16:23 UTC · model grok-4.3

classification 🧮 math.DS MSC 37H1537D2537A30
keywords Lyapunov exponentsrandom matrix productsMarkov shiftsuniform hyperbolicityprojective hyperbolicitypolynomial-time approximationanalytic dependenceinfinite matrices
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The pith

If 2x2 matrices are projectively uniformly hyperbolic under a Markov shift, the top Lyapunov exponent equals an explicit infinite-matrix sum that converges rapidly enough for a polynomial-time algorithm.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper shows that when a finite collection of invertible 2-by-2 matrices satisfies projective uniform hyperbolicity with respect to a transitive Markov shift, the top Lyapunov exponent for the associated random products admits a closed-form expression built from an infinite matrix. This representation converges fast enough that any prescribed accuracy can be reached with a number of arithmetic operations that grows only like the cube of the logarithm of the reciprocal error. The same hypothesis also implies that the exponent varies real-analytically with the matrix entries and the transition probabilities in a neighborhood of any such hyperbolic system, and that the coefficients of its Taylor series can themselves be computed efficiently. A reader cares because Lyapunov exponents quantify long-term growth in random or chaotic linear systems, and an explicit, rapidly evaluable formula replaces slow Monte-Carlo sampling with deterministic, high-precision arithmetic.

Core claim

For a finite family of invertible 2x2 real matrices and a transitive Markov shift on the index set, if the matrices are projectively uniformly hyperbolic with respect to the shift, then the top Lyapunov exponent λ admits an explicit representation in terms of an infinite matrix. This representation converges rapidly and yields a polynomial-time algorithm requiring only O((log(1/ε))³) arithmetic operations to achieve error ε. Moreover, λ depends real analytically on the matrix entries and transition probabilities near any projectively uniformly hyperbolic system, and each Taylor coefficient can be approximated in polynomial time.

What carries the argument

An infinite matrix whose entries are built from the projective actions of the given matrices along admissible Markov paths; the top Lyapunov exponent equals the logarithm of the spectral radius (or a related growth quantity) of this matrix.

If this is right

  • Any prescribed accuracy ε for λ can be reached with only O((log(1/ε))³) arithmetic operations.
  • Near a projectively uniformly hyperbolic system the exponent λ is a real-analytic function of both the matrix entries and the transition probabilities.
  • Every coefficient in the Taylor series of λ with respect to these parameters can be approximated to any fixed precision in polynomial time.
  • The same infinite-matrix construction supplies an explicit formula rather than an existence proof for the exponent.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The method may extend to computing other Lyapunov exponents or to the full Lyapunov spectrum once analogous hyperbolicity conditions are verified.
  • Because the representation is explicit and analytic, it could be differentiated under the sum to obtain derivatives of λ without additional Monte-Carlo sampling.
  • The polynomial-time guarantee suggests that the same technique might produce certified bounds on the exponent rather than merely numerical approximations.

Load-bearing premise

The given matrices must be projectively uniformly hyperbolic with respect to the transitive Markov shift.

What would settle it

For a concrete finite set of 2x2 matrices and a small Markov graph that satisfies projective uniform hyperbolicity, compute the first N terms of the infinite-matrix series for λ and compare the result to an independent long-run numerical estimate of the same exponent obtained by iterating random products; a persistent discrepancy larger than the truncation error would falsify the representation.

read the original abstract

We consider a finite family of invertible $2 \times 2$ real matrices and a transitive Markov shift on the index set. Let $\lambda$ 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 $\lambda$ admits an explicit representation in terms of an infinite matrix. This rapidly convergent representation yields a polynomial-time algorithm for approximating $\lambda$: only $O\big( (\log(1/\varepsilon))^3 \big)$ arithmetic operations are needed to achieve error $\varepsilon$. Furthermore, $\lambda$ 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The paper considers a finite family of invertible 2×2 real matrices together with a transitive Markov shift on the index set. Under the assumption that the matrices are projectively uniformly hyperbolic with respect to this shift, it proves that the top Lyapunov exponent λ admits an explicit representation as a functional of an infinite matrix constructed from the projective action and the Markov transition probabilities. The representation is rapidly convergent, yielding an algorithm that approximates λ to precision ε in O((log(1/ε))^3) arithmetic operations. The paper further shows that λ depends real-analytically on the matrix entries and transition probabilities in a neighborhood of any projectively uniformly hyperbolic system, with each Taylor coefficient itself approximable in polynomial time.

Significance. If the central claims hold, the work supplies both a theoretically explicit formula and a practical, rigorously analyzed algorithm for computing Lyapunov exponents in the uniformly hyperbolic regime, which is a setting of independent interest in random dynamical systems and ergodic theory. The polynomial-time complexity bound, the explicit dependence of all constants on the hyperbolicity data, and the analyticity result (with computable coefficients) constitute concrete strengths that go beyond existence statements.

minor comments (3)
  1. [§2.2] §2.2, Definition 2.3: the projective uniform hyperbolicity condition is stated in terms of a uniform contraction on the projective line, but the precise dependence of the constants C and θ on the Markov shift and the matrix family is not recorded explicitly; adding a short remark on how these constants enter the later error estimates would improve readability.
  2. [§4.1] §4.1, Eq. (4.2): the infinite matrix is defined via a block decomposition indexed by cylinder sets; the notation for the off-diagonal decay rate could be aligned more clearly with the hyperbolicity constants introduced in §2.3 to avoid any ambiguity when the reader compares the exponential decay claim with the complexity bound in Theorem 5.1.
  3. [§5.3] §5.3, Algorithm 5.2: the truncation procedure is described with explicit error bounds, but a brief remark on the arithmetic cost of matrix-vector multiplications for the truncated blocks (beyond the O((log(1/ε))^3) total) would make the complexity analysis fully self-contained.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful reading and positive assessment of our manuscript, including the detailed summary of the main results on the explicit representation of the top Lyapunov exponent, the O((log(1/ε))^3) approximation algorithm, and the real-analytic dependence. We accept the recommendation for minor revision and will incorporate any editorial or minor clarifications in the revised version.

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained from hyperbolicity

full rationale

The central claim derives an infinite-matrix representation for the top Lyapunov exponent directly from the projective uniform hyperbolicity assumption with respect to the Markov shift, using contraction properties on the projective line to obtain exponential off-diagonal decay and the stated O((log(1/ε))^3) complexity bound. No load-bearing step reduces by definition or construction to a fitted input, renamed empirical pattern, or self-citation chain; the argument remains conditional on the external hyperbolicity hypothesis and yields independent analytic dependence results. This is the expected non-circular outcome for a paper whose strongest claims are explicitly conditional.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the domain assumption of projective uniform hyperbolicity; no free parameters or invented entities are mentioned in the abstract.

axioms (1)
  • domain assumption Projective uniform hyperbolicity of the finite family of 2x2 matrices with respect to the transitive Markov shift
    This hypothesis is invoked to guarantee the existence of the infinite-matrix representation and all convergence and analyticity claims.

pith-pipeline@v0.9.0 · 5415 in / 1283 out tokens · 60103 ms · 2026-05-10T16:23:35.238367+00:00 · methodology

discussion (0)

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Reference graph

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