Skip to main content Accessibility help
×
Home

Dynamics and eigenvalues in dimension zero

  • LUIS HERNÁNDEZ-CORBATO (a1), DAVID JESÚS NIEVES-RIVERA (a2), FRANCISCO R. RUIZ DEL PORTAL (a2) and JAIME J. SÁNCHEZ-GABITES (a3)

Abstract

Let $X$ be a compact, metric and totally disconnected space and let $f:X\rightarrow X$ be a continuous map. We relate the eigenvalues of $f_{\ast }:\check{H}_{0}(X;\mathbb{C})\rightarrow \check{H}_{0}(X;\mathbb{C})$ to dynamical properties of $f$ , roughly showing that if the dynamics is complicated then every complex number of modulus different from 0, 1 is an eigenvalue. This stands in contrast with a classical inequality of Manning that bounds the entropy of $f$ below by the spectral radius of $f_{\ast }$ .

Copyright

References

Hide All
[1] Buescu, J., Kulczycki, M. and Stewart, I.. Liapunov stability and adding machines revisited. Dyn. Syst. 21 (2006), 379384.
[2] Buescu, J. and Stewart, I.. Liapunov stability and adding machines. Ergod. Th. & Dynam. Sys. 15 (1995), 271290.
[3] Hatcher, A.. Algebraic Topology. Cambridge University Press, Cambridge, 2002.
[4] Hurewicz, W. and Wallman, H.. Dimension Theory (Princeton Mathematical Series, 4) . Princeton University Press, Princeton, NJ, 1941.
[5] Manning, A.. Topological entropy and the first homology group. Dynamical Systems—Warwick 1974 (Proc. Symp. Appl. Topology and Dynamical Systems, University of Warwick, Coventry, 1973/1974) (Lecture Notes in Mathematics, 468) . Springer, Berlin, 1975, pp. 185190.
[6] Spanier, E. H.. Cohomology theory for general spaces. Ann. of Math. (2) 49 (1948), 407427.
[7] Spanier, E. H.. Algebraic Topology. McGraw-Hill, New York, 1966.

Keywords

MSC classification

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed