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GROWTH IN TOPOLOGICAL COMPLEXITY AND VOLUME GROWTH FOR RANDOM DYNAMICAL SYSTEMS

YUJUN ZHU

School of Mathematical Sciences, Peking University, Beijing 100871, P. R. China

College of Mathematics and Information Science, Hebei Normal University, Shijiazhuang 050016, P. R. China

https://doi.org/10.1142/S0219493706001827Cited by:3 (Source: Crossref)

Abstract

In this paper, relations between topological entropy, volume growth and the growth in topological complexity from homotopical and homological point of view are discussed for random dynamical systems. It is shown that, under certain conditions, the volume growth, the growth in fundamental group and the growth in homological group are all bounded from above by the topological entropy.

Keywords:

AMSC: 37B40, 37C35, 37H99

References

L. Arnold , Random Dynamical Systems ( Springer , 1998 ) . Crossref, Google Scholar
T. Bogenschutz, Random Comput. Dyn. 1, 99 (1992). Google Scholar
J. Franks and M. Misiurewicz, Handbook of Dynamical Systems, eds. B. Hasselblatt and A. Katok (Elsevier, 2005) pp. 547–598. Google Scholar
A. Katok and B. Hasselblatt , Introduction to the Modern Theory of Dynamical Systems ( Cambridge Univ. Press , 1995 ) . Crossref, Google Scholar
Y. Kifer , Ergodic Theory of Random Transformations ( Birkhäuser , 1986 ) . Crossref, Google Scholar
Y. Kifer and Y. Yomdin, Lecture Notes in Math. 1342 (Springer-Verlag, 1988) pp. 361–373. Google Scholar
Y. Kifer and P.-D. Liu, Handbook of Dynamical Systems 18, eds. B. Hasselblatt and A. Katok (Elsevier, 2006) pp. 379–499. Crossref, Google Scholar
P.-D. Liu, Ergo. Theor. Dyn. Syst. 21, 1279 (2001), DOI: 10.1017/S0143385701001614. Web of Science, Google Scholar
P.-D. Liu and M. Qian , Lect. Notes in Math. 1606 ( Springer , 1995 ) . Google Scholar
A. Manning, Lect. Notes in Math. 468 (Springer-Verlag, 1975) pp. 185–190. Google Scholar
M. Misiurewicz and F. Przytycki, Bull. Acad. Pol. Sci., Ser. Sci. Math., Astr. Phys. 25, 573 (1977). Web of Science, Google Scholar
M. Shub, Bull. Amer. Math. Soc. 80, 27 (1974). Crossref, Web of Science, Google Scholar
P. Walters , An Introduction to Ergodic Theory ( Springer-Verlag , 1982 ) . Crossref, Google Scholar
Y. Yomdin, Israel J. Math. 57, 285 (1987). Crossref, Web of Science, Google Scholar
Y. Yomdin, Israel J. Math. 57, 301 (1987). Crossref, Web of Science, Google Scholar
J.-L. Zhang, Y.-J. Zhu and L.-F. He, Acta Math. Sinica 48, 693 (2005). Google Scholar