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Convex dynamics with constant input

Published online by Cambridge University Press:  30 September 2009

R. L. ADLER ,
T. NOWICKI ,
G. ŚWIRSZCZ  and
C. TRESSER
R. L. ADLER
Affiliation:
IBM, TJ Watson Research Center, Yorktown Heights, NY 10598-0218, USA (email: [email protected], [email protected], [email protected], [email protected])
T. NOWICKI
Affiliation:
IBM, TJ Watson Research Center, Yorktown Heights, NY 10598-0218, USA (email: [email protected], [email protected], [email protected], [email protected])
G. ŚWIRSZCZ
Affiliation:
IBM, TJ Watson Research Center, Yorktown Heights, NY 10598-0218, USA (email: [email protected], [email protected], [email protected], [email protected])
C. TRESSER
Affiliation:
IBM, TJ Watson Research Center, Yorktown Heights, NY 10598-0218, USA (email: [email protected], [email protected], [email protected], [email protected])
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Abstract

In Adler et al [Convex dynamics and applications. Ergod. Th. & Dynam. Sys.25 (2005), 321–352] certain piecewise linear maps were defined in terms of a convex polytope. When the convex polytope is a simplex, the resulting map has a dual nature. On one hand it is defined on ℝN and acts as a piecewise translation. On the other it can be viewed as a translation on the N-torus. What relates its two roles? A natural answer would be that there exists an invariant fundamental set into which all orbits under piecewise translation eventually enter. We prove this for N=1 and for acute and right triangles—i.e. non-obtuse triangles. We leave open the case of obtuse triangles and higher-dimensional simplices. Another question not treated is the connectivity of the invariant fundamental sets which arise.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 30 , Issue 4 , August 2010 , pp. 957 - 972
Copyright
Copyright © Cambridge University Press 2009

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References

[1]Adler, R., Kitchens, B., Martens, M., Pugh, C., Shub, M. and Tresser, C.. Convex dynamics and applications. Ergod. Th. & Dynam. Sys. 25 (2005), 321–352.CrossRefGoogle Scholar
[2]Adler, R., Nowicki, T., Świrszcz, G., Tresser, C. and Winograd, S.. Convex dynamics: the lattices for the sub-tiles. (2008) in preparation.Google Scholar
[3]Güntürk, C. S. and Thao, N. T.. Ergodic dynamics in sigma–delta quantization: tiling invariant sets and spectral analysis of error. Adv. Appl. Math. 34(3) (2005), 523–560.CrossRefGoogle Scholar
[4]Hartshorne, R.. A Companion to Euclid’s Elements (Berkeley Mathematics Lecture Notes, 7). American Mathematical Society, Providence, RI, 1997.Google Scholar
[5]Tresser, C.. Bounding the errors for convex dynamics on one or more polytopes. Chaos 17 (2005), 033110.CrossRefGoogle Scholar