THOMAE’S FUNCTION AND THE SPACE OF ERGODIC MEASURES

The paper defines f: T^2 -> T^2 by f(x,y)=(x,x+y) and the time-average map T(ω)=lim_n (1/n)∑_{k=0}^{n-1} δ_{f^k(ω)} in the weak-* topology, and proves: (i) T(ω) is an ergodic invariant measure; (ii) T is continuous exactly at points with irrational first coordinate and discontinuous on a dense set; (iii) the image T(T^2) is homeomorphic to the surface of revolution R arising from Thomae’s function. These claims are stated and proved in Theorem 1 and Lemma 2 (using Weyl equidistribution for the irrational case and periodic averaging for the rational case), and a two-step homeomorphism construction via R̃ ⊂ S^1×C to R is sketched for (iii) . The model solution independently derives the same limits (via Fejér/Weyl-type arguments), the same continuity set, and a detailed homeomorphism to R using Fourier coefficients to prove bijectivity and two-sided continuity. Both are correct; the approaches differ mainly in technical tools and in how the homeomorphism is verified.