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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

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