On the geometry and regularity of invariant sets of piecewise-affine automorphisms on the Euclidean space

Abstract

<jats:p>In this paper, we derive geometric and analytic properties of invariant sets, including orbit closures, of a large class of piecewise-affine maps <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" mimetype="image" xlink:href="S0143385718001402_inline1.png" /><jats:tex-math>$T$</jats:tex-math></jats:alternatives></jats:inline-formula> on <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" mimetype="image" xlink:href="S0143385718001402_inline2.png" /><jats:tex-math>$\mathbb{R}^{d}$</jats:tex-math></jats:alternatives></jats:inline-formula>. We assume that (i) <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" mimetype="image" xlink:href="S0143385718001402_inline3.png" /><jats:tex-math>$T$</jats:tex-math></jats:alternatives></jats:inline-formula> consists of finitely many affine maps defined on a Borel measurable partition of <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" mimetype="image" xlink:href="S0143385718001402_inline4.png" /><jats:tex-math>$\mathbb{R}^{d}$</jats:tex-math></jats:alternatives></jats:inline-formula>, (ii) there is a lattice <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" mimetype="image" xlink:href="S0143385718001402_inline5.png" /><jats:tex-math>$\mathscr{L}\subset \mathbb{R}^{d}$</jats:tex-math></jats:alternatives></jats:inline-formula> that contains all of the mutual differences of the translation vectors of these affine maps, and (iii) all of the affine maps have the same linear part that is an automorphism of <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" mimetype="image" xlink:href="S0143385718001402_inline6.png" /><jats:tex-math>$\mathscr{L}$</jats:tex-math></jats:alternatives></jats:inline-formula>. We prove that finite-volume invariant sets of such piecewise-affine maps always consist of translational tiles relative to this lattice, up to some multiplicity. When the partition is Jordan measurable, we show that closures of bounded orbits of <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" mimetype="image" xlink:href="S0143385718001402_inline7.png" /><jats:tex-math>$T$</jats:tex-math></jats:alternatives></jats:inline-formula> are invariant and yield Jordan measurable tiles, again up to some multiplicity. In the latter case, we show that compact <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" mimetype="image" xlink:href="S0143385718001402_inline8.png" /><jats:tex-math>$T$</jats:tex-math></jats:alternatives></jats:inline-formula>-invariant sets also consist of Jordan measurable tiles. We then utilize these results to quantify the rate of convergence of ergodic averages for <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" mimetype="image" xlink:href="S0143385718001402_inline9.png" /><jats:tex-math>$T$</jats:tex-math></jats:alternatives></jats:inline-formula> in the case of bounded single tiles.</jats:p>