The Lyapunov exponent of a polynomial measures the average rate at which nearby orbits separate — the exponential sensitivity to initial conditions, averaged over the maximal entropy measure. For polynomials with disconnected Julia sets, the Lyapunov exponent encodes global dynamical information.
The paper on rigidity of Lyapunov exponents (arXiv: 2603.21179) proves that for polynomials in Q̄[z] (algebraic coefficients), equal Lyapunov exponents imply intertwining: f and g have the same exponent if and only if they are related by an affine conjugacy (f and g are intertwined, or f and ḡ are intertwined).
This is a rigidity result: a single number (the Lyapunov exponent) determines the polynomial up to a small equivalence class. The exponent is a real number derived from averaging over a fractal measure, yet it captures enough of the algebraic structure to distinguish non-conjugate polynomials.
An analogous result holds for critical heights — another global invariant measuring the escape rate of critical points. As an application, the multiplier spectrum morphism (which records the multipliers of all periodic orbits) is proved to be generically injective on the moduli space of degree-d polynomials.
The through-claim: a single averaged quantity determines the dynamics up to symmetry. The Lyapunov exponent is a coarse invariant — one number extracted from an infinite-dimensional dynamical system. But for polynomials with algebraic coefficients and disconnected Julia sets, this one number is (almost) a complete invariant. Coarseness and determination coexist when the algebraic structure is rigid enough.
2603.21179. Dynamical systems / Lyapunov exponents / polynomial dynamics / rigidity / multiplier spectrum.