Optimal Maps on the Torus, Uniquely Mapped Pairs and the Displacement Field of a Map

In the setting of The Flat Torus: Standing Notation, used with $n=d$ for a natural number $d$ with $1\le d$, and of Probability Measures on Euclidean Space and Random Vectors: Standing Notation, let $\varpi$ be the wrapped displacement, let $\mathcal{P}(\mathbb{T}^{d})$ be as in Probability Measures on the Flat Torus, the Torus Cost of a Coupling, the Torus Wasserstein Distance and Optimal Couplings §measures and optimality of a coupling as defined there, let $\mu,\nu\in\mathcal{P}(\mathbb{T}^{d})$, and write $\mathrm{id}$ for the identity map of $\mathbb{R}^{d}$. For a Borel map $T:\mathbb{R}^{d}\to\mathbb{R}^{d}$ with $T_{\#}\mu=\nu$ the push-forward $(\mathrm{id},T)_{\#}\mu$ belongs to $\Pi(\mu,\nu)$ by Couplings on Euclidean Space: Product Coupling, Swap, Finiteness of the Cost, Push-Forward Couplings, Modifying One Marginal, Quantisation, Gluing over a Finitely Supported Measure, and the Lipschitz Bound §pushforward.

1. (Optimal map)¶ A Borel map $T:\mathbb{R}^{d}\to\mathbb{R}^{d}$ is an optimal map on the torus from $\mu$ to $\nu$ if $T_{\#}\mu=\nu$ and $(\mathrm{id},T)_{\#}\mu$ is optimal.

2. (Uniquely mapped pair)¶ The ordered pair $(\mu,\nu)$ is uniquely mapped on the torus if there is an optimal map $T$ on the torus from $\mu$ to $\nu$ such that every optimal coupling of $\mu$ and $\nu$ equals $(\mathrm{id},T)_{\#}\mu$.

3. (Displacement field)¶ For a Borel map $T:\mathbb{R}^{d}\to\mathbb{R}^{d}$, the displacement field of $T$ is the map $v_{T}:\mathbb{R}^{d}\to\mathbb{R}^{d}$, $v_{T}(x)=\varpi\bigl(T(x)-x\bigr)$. It is Borel, as the composite of the Borel map $\varpi$ of The Flat Torus Distance: Minimality of the Wrapped Displacement, Periodicity, the Metric on the Unit Cell and the Lipschitz Bound §lipschitz with $T-\mathrm{id}$, whose components are differences of Borel functions (claim 2 of The Borel Sigma-Algebra of a Euclidean Space as a Product, and Measurability of Projections, Sequentially Continuous Maps, and Open and Closed Sets and claim 2 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions).