*Published Paper*

**Inserted:** 13 jul 2023

**Last Updated:** 13 jul 2023

**Journal:** Math. Z.

**Volume:** 284

**Number:** 3-4

**Pages:** 1021–1034

**Year:** 2016

**Abstract:**

We derive a discrete version of the results of 2. If $M$ is a compact metric space, $c : M\times M \to \R$ a continuous cost function and $\lambda \in (0,1)$, the unique solution to the discrete $\lambda$-discounted equation is the only function $u_\lambda : M\to \R$ such that \[ \forall x\in M, \quad u_\lambda(x) = \min_{y\in M} \lambda u_\lambda (y) + c(y,x). \] We prove that there exists a unique constant $\alpha\in \R$ such that the family of $u_\lambda+\alpha/(1-\lambda)$ is bounded as $\lambda \to 1$ and that for this $\alpha$, the family uniformly converges to a function $u_0 : M\to \R$ which then verifies \[ \forall x\in X, \quad u_0(x) = \min_{y\in X}u_0(y) + c(y,x)+\alpha. \] The proofs make use of Discrete Weak KAM theory. We also characterize $u_0$ in terms of Peierls barrier and projected Mather measures.

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