Ergodic and Entropic Behavior of the Harmonic Map Heat Flow to the Moduli Space of Flat Tori

We investigate the harmonic map heat flow from a compact Riemannian manifold \( M \) into the moduli space \( \mathcal{M}_1 \) of unit-area flat tori, which carries a natural hyperbolic structure as the quotient \( \mathrm{SL}(2,\mathbb{Z}) \backslash \mathbb{H} \). We prove that the flow is stable with respect to the energy functional and exhibits ergodic behavior in the sense that the evolving maps asymptotically distribute their image uniformly across the moduli space. As a concrete contribution, we show that the sequence of pushforward measures under the flow converges weak--$^{*}$ to the normalized hyperbolic measure on \( \mathcal{M}_1 \). Moreover, we introduce a relative entropy framework to measure the statistical deviation of the flow from equilibrium and prove that the relative entropy with respect to the hyperbolic measure decays to zero in the long-time limit. This provides a quantitative refinement of the ergodic result and establishes a connection between geometric flows, moduli space dynamics, and information-theoretic convergence.