# American Institute of Mathematical Sciences

March  2003, 9(2): 323-338. doi: 10.3934/dcds.2003.9.323

## Quasi-invariant attractors of piecewise isometric systems

 1 Department of Mathematics and statistics, University of Surrey, Guidford GU2 7XH, United Kingdom

Received  July 2001 Revised  March 2002 Published  December 2002

We describe new examples of global attractors arising in planar piecewise rotations with two convex atoms. The dynamics inside these attractors is proved to be equivalent to that from models of digital filters.
We also discuss some subtleties on the definition of piecewise isometric attractors and their properties, motivated not only by our new examples but also by others existing in the literature. More precisely, we require a minimality condition so that uniqueness is guaranteed and also, we establish equivalent forms of attractivity under some regularity assumption. The notion of quasi-invariance is introduced as it proves to be a suitable concept in the context of planar piecewise rotations.
Citation: Miguel Ângelo De Sousa Mendes. Quasi-invariant attractors of piecewise isometric systems. Discrete and Continuous Dynamical Systems, 2003, 9 (2) : 323-338. doi: 10.3934/dcds.2003.9.323
 [1] Artem Dudko. Computability of the Julia set. Nonrecurrent critical orbits. Discrete and Continuous Dynamical Systems, 2014, 34 (7) : 2751-2778. doi: 10.3934/dcds.2014.34.2751 [2] Marcello Trovati, Peter Ashwin, Nigel Byott. Packings induced by piecewise isometries cannot contain the arbelos. Discrete and Continuous Dynamical Systems, 2008, 22 (3) : 791-806. doi: 10.3934/dcds.2008.22.791 [3] Jeremy LeCrone, Gieri Simonett. Continuous maximal regularity and analytic semigroups. Conference Publications, 2011, 2011 (Special) : 963-970. doi: 10.3934/proc.2011.2011.963 [4] Pascal Auscher, Sylvie Monniaux, Pierre Portal. The maximal regularity operator on tent spaces. Communications on Pure and Applied Analysis, 2012, 11 (6) : 2213-2219. doi: 10.3934/cpaa.2012.11.2213 [5] Boris Hasselblatt. Critical regularity of invariant foliations. Discrete and Continuous Dynamical Systems, 2002, 8 (4) : 931-937. doi: 10.3934/dcds.2002.8.931 [6] Fritz Colonius. Invariance entropy, quasi-stationary measures and control sets. Discrete and Continuous Dynamical Systems, 2018, 38 (4) : 2093-2123. doi: 10.3934/dcds.2018086 [7] Yoshikazu Giga, Jürgen Saal. $L^1$ maximal regularity for the laplacian and applications. Conference Publications, 2011, 2011 (Special) : 495-504. doi: 10.3934/proc.2011.2011.495 [8] Jeremy LeCrone, Gieri Simonett. On quasilinear parabolic equations and continuous maximal regularity. Evolution Equations and Control Theory, 2020, 9 (1) : 61-86. doi: 10.3934/eect.2020017 [9] Yuanzhen Shao. Continuous maximal regularity on singular manifolds and its applications. Evolution Equations and Control Theory, 2016, 5 (2) : 303-335. doi: 10.3934/eect.2016006 [10] Mark F. Demers. Uniqueness and exponential mixing for the measure of maximal entropy for piecewise hyperbolic maps. Discrete and Continuous Dynamical Systems, 2021, 41 (1) : 217-256. doi: 10.3934/dcds.2020217 [11] Michihiro Hirayama. Periodic probability measures are dense in the set of invariant measures. Discrete and Continuous Dynamical Systems, 2003, 9 (5) : 1185-1192. doi: 10.3934/dcds.2003.9.1185 [12] Peng Huang, Xiong Li, Bin Liu. Invariant curves of smooth quasi-periodic mappings. Discrete and Continuous Dynamical Systems, 2018, 38 (1) : 131-154. doi: 10.3934/dcds.2018006 [13] Yannick Privat, Emmanuel Trélat, Enrique Zuazua. Complexity and regularity of maximal energy domains for the wave equation with fixed initial data. Discrete and Continuous Dynamical Systems, 2015, 35 (12) : 6133-6153. doi: 10.3934/dcds.2015.35.6133 [14] Carlos Lizama, Marina Murillo-Arcila. Discrete maximal regularity for volterra equations and nonlocal time-stepping schemes. Discrete and Continuous Dynamical Systems, 2020, 40 (1) : 509-528. doi: 10.3934/dcds.2020020 [15] Irena Lasiecka, Mathias Wilke. Maximal regularity and global existence of solutions to a quasilinear thermoelastic plate system. Discrete and Continuous Dynamical Systems, 2013, 33 (11&12) : 5189-5202. doi: 10.3934/dcds.2013.33.5189 [16] Victoriano Carmona, Emilio Freire, Soledad Fernández-García. Periodic orbits and invariant cones in three-dimensional piecewise linear systems. Discrete and Continuous Dynamical Systems, 2015, 35 (1) : 59-72. doi: 10.3934/dcds.2015.35.59 [17] Hebai Chen, Jaume Llibre, Yilei Tang. Centers of discontinuous piecewise smooth quasi–homogeneous polynomial differential systems. Discrete and Continuous Dynamical Systems - B, 2019, 24 (12) : 6495-6509. doi: 10.3934/dcdsb.2019150 [18] Heide Gluesing-Luerssen. On isometries for convolutional codes. Advances in Mathematics of Communications, 2009, 3 (2) : 179-203. doi: 10.3934/amc.2009.3.179 [19] Serhii Dyshko. On extendability of additive code isometries. Advances in Mathematics of Communications, 2016, 10 (1) : 45-52. doi: 10.3934/amc.2016.10.45 [20] Nils Ackermann, Thomas Bartsch, Petr Kaplický. An invariant set generated by the domain topology for parabolic semiflows with small diffusion. Discrete and Continuous Dynamical Systems, 2007, 18 (4) : 613-626. doi: 10.3934/dcds.2007.18.613

2021 Impact Factor: 1.588