Nonunique minimizers allow distinct local accumulation limits

Let $P,R$ be stationary laws of finite specific entropy and let $P_{\theta }=\theta P+(1-\theta )R$. For a bounded cube $G$, the binary-label chain rule [F3] gives $\theta D(P_G\mathrm{\parallel }{Q}_G)+(1-\theta )D(R_G\mathrm{\parallel }{Q}_G)=D((P_{\theta }{)}_G\mathrm{\parallel }{Q}_G)+I(S;X_G),$ where $S$ is the mixture label and $X_G$ the observed configuration. Since $0\le I(S;X_G)\le H(S)<\mathrm{\infty }$, divide by $\mathrm{\mid }G\mathrm{\mid }$ and take the cube limit [F2]; $H(S)\mathrm{/}\mathrm{\mid }G\mathrm{\mid }\to 0$, so $I(P_{\theta })=\theta I(P)+(1-\theta )I(R)$. The expectation defining $\rho$ is affine. For finite energy densities, $E_{P_{\theta }}{H}_G=\theta E_P{H}_G+(1-\theta )E_R{H}_G$ for every $G$; dividing by volume and taking limits gives affinity of $\mathrm{\Phi }$. Thus $J$ is affine on finite mixtures of finite-$J$ laws.

Applying step 1.1 to $P_0,P_1$ gives $J(P_{\theta })=\theta J(P_0)+(1-\theta )J(P_1)=m$, so every mixture is a minimizer. Put $P_{2n}=P_0$ and $P_{2n+1}=P_1$. The even and odd subsequences converge locally to $P_0$ and $P_1$. Since distinct laws differ on a bounded-region measurable event, its indicator is a bounded local test by [F4] with different expectations; the full sequence therefore has no ${\tau }_L$ limit. Both accumulation points are minimizers.