Thick Arnold tongues

The paper constructs, for the inertial particle system x¨=-(1/ε)(x˙-v(x)) with v=J∇H and H(x,y)=cos x cos y - a x + b y, a global attracting 2D invariant manifold and a Poincaré section on the torus, reducing the dynamics to a monotone one-parameter family of circle maps with two flat spots. It proves: (i) existence of a common drift slope for almost all initial data; (ii) Cantor-like dependence of the drift slope on α=a/b with plateaus at rationals; (iii) zero Hausdorff dimension of the irrational set; and (iv) exponential “steepness to all orders” at plateau endpoints. Key ingredients include an explicit first-order reduction ẋ=v+εf with f=−(Dv)v+O(ε), a contraction estimate for the forward Poincaré map via negative time-averaged divergence, and a monotonicity/expansivity framework establishing Theorem 1.3 for circle maps with flat spots. These address precisely the obstacles the model deemed unresolved. See the paper’s Proposition 3.1 (Fenichel reduction) and explicit f (and div f) formulae, Proposition 3.2 (phase portrait, section, and monotone family Qs), the contraction estimate for P, the rotation-number-to-drift link, and Theorem 1.1 enumerating items (1)–(4) with their proofs via Theorem 1.3.