The growth rate hidden in random matrix products
Take a finite family of invertible non-negative 2×2 matrices with positive weights summing to one, and draw an i.i.d. sequence of them. The top Lyapunov exponent λ is the almost sure limit of (1/n) log ‖A_in ⋯ A_i1‖ as n grows, and it does not depend on the chosen norm. In words, λ is the exponential rate at which a random product stretches a vector.
Existence of this limit is classical, but an explicit, fast-converging expression for λ was still difficult even in the 2×2 case. The paper's motivating application is dimension theory: λ appears when computing the Hausdorff dimension of a Cantor-type fractal intersected with a random translate of itself. That link is what makes a computable formula for λ more than a numerical convenience.
Kernel expansion replaces enumerating all products
Earlier work by Pollicott represented λ through the root of a Fredholm determinant of a transfer operator. Its n-th coefficient requires computing every product of matrices of length n, so the number of products grows combinatorially; for 2×2 matrices the resulting cost is subexponential, needing O(exp(α√log(1/ε))) arithmetic operations for truncation error ε.
This paper instead expands the integrand of the Furstenberg–Kifer variational formula in a family of functions {1} ∪ {φ_n}, where each φ_n integrates to zero against every stationary measure of the transposed Möbius maps. The author calls this Kernel expansion. The expansion produces the infinite matrix T and vector v in the series λ = Σ_{n≥0} (T^n v)_0, and the paper proves that T maps the relevant vector space V into itself, which is not apparent from the definition.
Truncation error falls with polynomial cost
Theorem 1.2 makes the series usable. For any ε > 0 there are integers N = O(log(1/ε)) and M = O(log(N/ε)) such that replacing T by its upper-left M×M submatrix T_M and v by its first M components leaves an error below ε. The total arithmetic cost is O((log(1/ε))^3).
The proof splits the error into two parts. The tail of the Neumann series is controlled by a Wasserstein contraction of the stationary-measure operator H, giving a geometric bound E r^{n−1} with contraction rate r < 1. The truncation error is harder: termwise absolute bounds blow up as the index k grows, so the paper derives contour-integral representations for T and T_M in which oscillatory cancellations become visible, then lifts them to iterates and closes the estimate with a telescoping identity.
Non-negative families qualify when no heteroclinic connection appears
The series is first proved for entrywise positive matrices. To extend it, the paper defines a heteroclinic connection as a matrix mapping an unstable direction of one matrix onto a stable direction of another. Theorem 1.4 proves three conditions equivalent: the family {A_i} ∪ {A_i A_j} has no generalized heteroclinic connections of depth 2; the projective Möbius maps admit an invariant closed arc; and there is an invertible P such that every P A_i P^{-1} is positive.
The third condition is what transfers the result: products conjugate by P, so the Lyapunov exponent of the original family equals that of the positive family, and Theorem 1.1 applies. The equivalence is sharp in a specific sense. Remark 1.5 notes that when generalized heteroclinic connections of depth 2 are present, the formal partial sums of the series may converge to the true value or may oscillate and fail to converge; Example 5.7 exhibits the failure.
Cantor set intersections inherit the series formula
Kenyon and Peres expressed dim_H((K1 + t) ∩ K2) for b-adic Cantor sets as λ / log b, where λ is the Lyapunov exponent of b matrices A_0, …, A_{b−1} built from the digit sets, provided no finite product of those matrices is zero. The new series therefore yields a computable dimension formula whenever the matrices can be conjugated to positive ones.
Proposition 1.8 covers the one-digit forbidden family: for b ≥ 5, D1 and D2 each omit a single digit τ and u. The pair is non-degenerate exactly when neither omitted digit is 0 or b−1 and τ + u = b−1; in that case there are no generalized heteroclinic connections of depth 2, so the series applies. For the middle-fifth Cantor set with D1 = D2 = {0,1,3,4}, the method gives a rigorously certified λ = 1.15935795532718828347… to 128 decimal places, hence dim_H(C5 ∩ (C5 + t)) = λ / log 5 for Lebesgue-a.e. t.
The coverage is not complete. The cases b = 4 with D1 = {0,1,3}, D2 = {0,2,3} and its mirror are non-degenerate but contain a parabolic map, producing generalized heteroclinic connections of depth 2; the paper states these are the only pairs in the one-digit forbidden family still beyond its reach. For thick Cantor sets, Proposition 1.12 shows that if each digit set keeps at least b − ⌊(b−3)/5⌋ digits and b ≥ 3, the pair is either degenerate or admits an invariant arc, so the series representation holds.
Three-term recurrences get an almost sure growth rate
The same machinery determines the almost sure growth rate of i.i.d. three-term recurrences x_{n+1} = a_{i_{n+1}} x_n + b_{i_{n+1}} x_{n−1}, starting from x_0 = x_1 = 1, where each pair (a_i, b_i) of positive numbers is drawn independently from a finite set with weights w_i. This extends the classical random Fibonacci setting, in which f_n = ±f_{n−1} ± f_{n−2} with independent uniform signs and the growth rate is 1.13198824…, to finitely many positive coefficient pairs.
The quantitative bounds remain a 2×2 result
The paper is explicit about scope. The algebraic, telescoping part of the Kernel-expansion argument extends to invertible positive (d+1)×(d+1) matrices, and the author sketches the d-simplex setup with the Hilbert projective metric and a multivariate version of the matrix T. But the quantitative truncation estimates behind the O((log(1/ε))^3) complexity are currently specific to 2×2 matrices, so the polynomial-cost guarantee should not be assumed in higher dimension.
A second boundary is the heteroclinic condition itself. The series representation for non-negative families depends on the absence of generalized heteroclinic connections of depth 2, and the paper leaves open when a digit pair is degenerate and, if non-degenerate, when such a connection occurs (Question 1.11). The practical priority follows from these two boundaries: the certified, fast-converging computation is available for positive 2×2 families and for non-negative families that pass the invariant-arc test, and the Cantor-set and recurrence applications inherit exactly that reach.
Sources used in this answer
Lyapunov exponents for random products of non-negative matrices
Proves an explicit, rapidly convergent Neumann-series representation for the top Lyapunov exponent of i.i.d. products of invertible positive 2×2 matrices, approximable with O((log(1/ε))^3) arithmetic operations, and extends it to non-negative families without generalized heteroclinic connections of depth 2.