TY - JOUR

T1 - On the geometry and regularity of invariant sets of piecewise-affine automorphisms on the Euclidean space

AU - Sinan Gunturk, C.

AU - Thao, Nguyen T.

N1 - Publisher Copyright:
© 2019 Cambridge University Press.

PY - 2020/8/1

Y1 - 2020/8/1

N2 - In this paper, we derive geometric and analytic properties of invariant sets, including orbit closures, of a large class of piecewise-affine maps on. We assume that (i) consists of finitely many affine maps defined on a Borel measurable partition of, (ii) there is a lattice 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. 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 are invariant and yield Jordan measurable tiles, again up to some multiplicity. In the latter case, we show that compact -invariant sets also consist of Jordan measurable tiles. We then utilize these results to quantify the rate of convergence of ergodic averages for in the case of bounded single tiles.

AB - In this paper, we derive geometric and analytic properties of invariant sets, including orbit closures, of a large class of piecewise-affine maps on. We assume that (i) consists of finitely many affine maps defined on a Borel measurable partition of, (ii) there is a lattice 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. 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 are invariant and yield Jordan measurable tiles, again up to some multiplicity. In the latter case, we show that compact -invariant sets also consist of Jordan measurable tiles. We then utilize these results to quantify the rate of convergence of ergodic averages for in the case of bounded single tiles.

KW - Arithmetic and algebraic dynamics

KW - Low-dimensional dynamics

KW - Tilings

KW - Topological dynamics

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U2 - 10.1017/etds.2018.140

DO - 10.1017/etds.2018.140

M3 - Article

AN - SCOPUS:85059605627

SN - 0143-3857

VL - 40

SP - 2183

EP - 2218

JO - Ergodic Theory and Dynamical Systems

JF - Ergodic Theory and Dynamical Systems

IS - 8

ER -