ERGODICITY IN PLANAR SLOW-FAST SYSTEMS THROUGH SLOW RELATION FUNCTIONS

The paper’s Theorem 1 classifies invariant measures of the slow relation function S in terms of zeros of the slow divergence integral Ĩ, using that S is continuous and increasing, the equivalence Ĩ(s)=0 ⇔ S(s)=s (from equation (31)), monotone convergence of iterates, and Poincaré recurrence to confine invariant measures to fixed points . The candidate’s solution proves the same classification via a different route: it constructs p(s)=limn→∞S^n(s) and shows μ=p_*μ for any invariant μ by dominated convergence, hence μ is supported on Fix(S), and then identifies Fix(S) with {0} ∪ {zeros of Ĩ}. This avoids explicit use of recurrence but relies on the same monotonicity/continuity properties of S that the paper establishes . The statements (A)–(C) match exactly Theorem 1’s parts (1)–(3).