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

Part of: Probabilistic theory: distribution modulo $1$; metric theory of algorithms

Published online by Cambridge University Press:  07 January 2019

C. SİNAN GÜNTÜRK  and
NGUYEN T. THAO
C. SİNAN GÜNTÜRK
Affiliation:
Courant Institute of Mathematical Sciences, NYU, 251 Mercer Street, New York, NY 10012, USA email [email protected]
NGUYEN T. THAO
Affiliation:
City College of New York, CUNY, Convent Avenue at 138th Street, New York, NY 10031, USA email [email protected]
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Abstract

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

Keywords

arithmetic and algebraic dynamicslow-dimensional dynamicstilingstopological dynamics

MSC classification

Secondary: 11K38: Irregularities of distribution, discrepancy
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 40 , Issue 8 , August 2020 , pp. 2183 - 2218
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Copyright
© Cambridge University Press, 2019

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