Hostname: page-component-586b7cd67f-g8jcs Total loading time: 0 Render date: 2024-11-20T15:27:35.387Z Has data issue: false hasContentIssue false
  • English
  • Français

Reduced dynamical systems

Part of: Complex dynamical systems Ergodic theory

Published online by Cambridge University Press:  26 May 2020

LUKA BOC THALER  and
UROŠ KUZMAN
LUKA BOC THALER
Affiliation:
Dipartimento Di Matematica, Università di Roma ‘Tor Vergata’, Italy email [email protected]
UROŠ KUZMAN
Affiliation:
Faculty of Mathematics and Physics, University of Ljubljana, Slovenia Institute of Mathematics, Physics and Mechanics, Ljubljana, Slovenia email [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We consider the dynamics of complex rational maps on $\widehat{\mathbb{C}}$. We prove that, after reducing their orbits to a fixed number of positive values representing the Fubini–Study distances between finitely many initial elements of the orbit and the origin, ergodic properties of the rational map are preserved.

Keywords

low dimensional dynamicsclassical ergodic theoryrational functions

MSC classification

Primary: 37F10: Polynomials; rational maps; entire and meromorphic functions 37A35: Entropy and other invariants, isomorphism, classification
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 41 , Issue 6 , June 2021 , pp. 1612 - 1626
Check for updates
Copyright
© The Author(s) 2020. Published by Cambridge University Press

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

Bergweiler, W. and Eremenko, A.. Meromorphic functions with linearly distributed values and julia sets of rational functions. Proc. Amer. Math. Soc. 137(7) (2009), 2329–2333.CrossRefGoogle Scholar
Bierstone, E. and Milman, P. D.. Semianalytic and subanalytic sets. Publ. Math. Inst. Hautes Études Sci. 67 (1988), 5–42.CrossRefGoogle Scholar
Boc Thaler, L.. A reconstruction theorem for complex polynomials. Internat. J. Math. 26(9) (2015), 1550073.CrossRefGoogle Scholar
Bowen, R.. Topological entropy for noncompact sets. Trans. Amer. Math. Soc. 184 (1973), 125–136.CrossRefGoogle Scholar
Downarowicz, T.. Entropy in Dynamical Systems (New Mathematical Monographs, 18) . Cambridge University Press, Cambridge, 2011.CrossRefGoogle Scholar
Downarowicz, T.. Positive topological entropy implies chaos DC2. Proc. Amer. Math. Soc. 142 (2014), 137–149.CrossRefGoogle Scholar
Eremenko, A. and van Strien, S.. Rational maps with real multipliers. Trans. Amer. Math. Soc. 363(12) (2011), 6453–6463.CrossRefGoogle Scholar
Fornæss, J. E. and Peters, H.. Complex dynamics with focus on the real part. Ergod. Th. & Dynam. Sys. 37(1) (2017), 176–192.CrossRefGoogle Scholar
Frisch, J.. Points de platitude d’un morphisme d’espaces analytiques complexes. Invent. Math. 4 (1967), 118–138.CrossRefGoogle Scholar
Hironaka, H.. Local analytic dimensions of a subanalytic set. Proc. Japan Acad. Ser. A Math. Sci. 62(2) (1986), 73–75.Google Scholar
Łojasiewicz, S.. On semi-analytic and subanalytic geometry. Panoramas of Mathematics (Warsaw, 1992/1994) (Banach Center Publications, 34) . Institute of Mathematics of the Polish Academy of Sciences, Warsaw, 1995, pp. 89–104.Google Scholar
Lyubich, M.. Entropy properties of rational endomorphisms of the riemann sphere. Ergod. Th. & Dynam. Sys. 898(3) (1983), 351–385.CrossRefGoogle Scholar
Mañé, R.. On the uniqueness of the maximizing measure for rational maps. Bol. Soc. Bras. Mat. 14 (1983), 27–43.CrossRefGoogle Scholar
Ruiz, J. M.. On Hilbert’s 17th problem and real Nullstellensatz for global analytic functions. Math. Z. 190(3) (1985), 447–454.CrossRefGoogle Scholar
Tamm, M.. Subanalytic sets in the calculus of variation. Acta Math. 146 (1981), 167–199.CrossRefGoogle Scholar