scholarly journals Dynamics of 2-interval piecewise affine maps and Hecke-Mahler series

2021 ◽  
Vol 17 (0) ◽  
pp. 33-63
Author(s):  
Michel Laurent ◽  
◽  
Arnaldo Nogueira
Keyword(s):  
Piecewise Affine ◽  
Affine Maps

On the preservation of properties by piecewise affine maps of locally compact groups

2019 ◽  
Vol 12 (3) ◽  
pp. 491-502
Author(s):  
Serina Camungol ◽  
Matthew Morison ◽  
Skylar Nicol ◽  
Ross Stokke
Keyword(s):  
Locally Compact Groups ◽  
Compact Groups ◽  
Locally Compact ◽  
Piecewise Affine ◽  
Affine Maps

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

2019 ◽  
Vol 40 (8) ◽  
pp. 2183-2218
Author(s):  
C. SİNAN GÜNTÜRK ◽  
NGUYEN T. THAO
Keyword(s):  
Euclidean Space ◽  
Rate Of Convergence ◽  
Linear Part ◽  
Invariant Sets ◽  
Measurable Partition ◽  
Ergodic Averages ◽  
Piecewise Affine ◽  
Affine Maps ◽  
Orbit Closures ◽  
Borel Measurable

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.

scholarly journals REACHABILITY PROBLEMS IN LOW-DIMENSIONAL ITERATIVE MAPS

2008 ◽  
Vol 19 (04) ◽  
pp. 935-951 ◽  
Author(s):  
OLEKSIY KURGANSKYY ◽  
IGOR POTAPOV ◽  
FERNANDO SANCHO-CAPARRINI
Keyword(s):  
Reachability Problem ◽  
One Dimensional ◽  
Piecewise Affine ◽  
Affine Maps ◽  
Iterative Maps ◽  
Low Dimensional

In this paper we analyze the dynamics of one-dimensional piecewise maps. We show that one-dimensional piecewise affine maps are equivalent to pseudo-billiard or so called “strange billiard” systems. We also show that use of more general classes of functions lead to undecidability of reachability problem for one-dimensional piecewise maps.

scholarly journals Completely bounded homomorphisms of the Fourier algebra revisited

2022 ◽  
Vol 0 (0) ◽  
Author(s):  
Matthew Daws
Keyword(s):  
Locally Compact Groups ◽  
Compact Groups ◽  
Fourier Algebra ◽  
Locally Compact ◽  
Piecewise Affine ◽  
Affine Maps

Abstract Assume that A ⁢ ( G ) A(G) and B ⁢ ( H ) B(H) are the Fourier and Fourier–Stieltjes algebras of locally compact groups 𝐺 and 𝐻, respectively. Ilie and Spronk have shown that continuous piecewise affine maps α : Y ⊆ H → G \alpha\colon Y\subseteq H\to G induce completely bounded homomorphisms Φ : A ⁢ ( G ) → B ⁢ ( H ) \Phi\colon A(G)\to B(H) and that, when 𝐺 is amenable, every completely bounded homomorphism arises in this way. This generalised work of Cohen in the abelian setting. We believe that there is a gap in a key lemma of the existing argument, which we do not see how to repair. We present here a different strategy to show the result, which instead of using topological arguments, is more combinatorial and makes use of measure-theoretic ideas, following more closely the original ideas of Cohen.

scholarly journals Jump-induced mixed-mode oscillations through piecewise-affine maps

2021 ◽  
pp. 125641
Author(s):  
Yiorgos Patsios ◽  
Renato Huzak ◽  
Peter De Maesschalck ◽  
Nikola Popović
Keyword(s):  
Mixed Mode ◽  
Piecewise Affine ◽  
Mixed Mode Oscillations ◽  
Affine Maps

scholarly journals Absolutely continuous invariant measures for some piecewise hyperbolic affine maps

2008 ◽  
Vol 28 (1) ◽  
pp. 211-228 ◽  
Author(s):  
TOMAS PERSSON
Keyword(s):  
Invariant Measure ◽  
Invariant Measures ◽  
Bounded Subset ◽  
Continuous Invariant Measure ◽  
Absolutely Continuous ◽  
Absolutely Continuous Invariant Measures ◽  
Piecewise Affine ◽  
Affine Maps ◽  
Absolutely Continuous Invariant Measure ◽  
Absolutely Continuous Invariant

AbstractA class of piecewise affine hyperbolic maps on a bounded subset of the plane is considered. It is shown that if a map from this class is sufficiently area-expanding then almost surely this map has an absolutely continuous invariant measure.

scholarly journals Chaotic sliding dynamics for planar piecewise affine maps

2017 ◽  
Author(s):  
Claudio Buzzi ◽  
João Medrado ◽  
Paulo Da Silva
Keyword(s):  
Piecewise Affine ◽  
Affine Maps ◽  
Sliding Dynamics

scholarly journals Topological dynamics of piecewise -affine maps

2016 ◽  
Vol 38 (5) ◽  
pp. 1876-1893 ◽  
Author(s):  
ARNALDO NOGUEIRA ◽  
BENITO PIRES ◽  
RAFAEL A. ROSALES
Keyword(s):  
Periodic Orbit ◽  
Periodic Orbits ◽  
Topological Dynamics ◽  
Limit Set ◽  
Real Numbers ◽  
Piecewise Affine ◽  
Affine Maps ◽  
Asymptotically Periodic

Let $-1<\unicode[STIX]{x1D706}<1$ and let $f:[0,1)\rightarrow \mathbb{R}$ be a piecewise $\unicode[STIX]{x1D706}$-affine contraction: that is, let there exist points $0=c_{0}<c_{1}<\cdots <c_{n-1}<c_{n}=1$ and real numbers $b_{1},\ldots ,b_{n}$ such that $f(x)=\unicode[STIX]{x1D706}x+b_{i}$ for every $x\in [c_{i-1},c_{i})$. We prove that, for Lebesgue almost every $\unicode[STIX]{x1D6FF}\in \mathbb{R}$, the map $f_{\unicode[STIX]{x1D6FF}}=f+\unicode[STIX]{x1D6FF}\,(\text{mod}\,1)$ is asymptotically periodic. More precisely, $f_{\unicode[STIX]{x1D6FF}}$ has at most $n+1$ periodic orbits and the $\unicode[STIX]{x1D714}$-limit set of every $x\in [0,1)$ is a periodic orbit.

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