Natural extensions for piecewise affine maps via Hofbauer towers (Q405734)

The authors present a method for obtaining some geometric natural extensions of piecewise continuous maps with locally constant Jacobian. More precisely, let \(X\) be a compact subset of \({\mathbb{R}^n},\) \({\mathcal{B}}\) be the Lebesgue \(\sigma\)-algebra on \(X\), \(\mu\) be a probability measure on \({(X,\mathcal{B})}\) and consider a map \({T:X\to X}\). It is assumed that this system have a partition with additional properties. Using this partition the authors define new system (Hofbauer tower) \({(\hat{X},\hat{\mathcal{B}})}\), with a map \({\hat{T}:\hat{X}\to\hat{X}}\) and some measures \(\hat{\mu}_n\) on \({(\hat{X},\hat{\mathcal{B}})}\). If the sequence \({\hat{\mu}_n}\) converges in the weak topology to a non-zero measure~\({\hat{\mu}}\), then for system \({(X,\mathcal{B}, \hat{\mu}\circ\pi^{-1}, T)}\) the authors construct the natural extension \({(Y,\mathcal{C},\nu, F)},\) where \(\pi:\hat{X}\to{X}\) is canonical projection and \(Y\) consists of pieces which extend the pieces of \(\hat{X}\) by one dimension. At the end of paper, some examples of transformations are considered.