Hostname: page-component-cd9895bd7-gxg78 Total loading time: 0 Render date: 2024-12-26T00:31:35.960Z Has data issue: false hasContentIssue false
  • English
  • Français

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])
Get access Rights & Permissions [Opens in a new window]

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

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