Hostname: page-component-586b7cd67f-vdxz6 Total loading time: 0 Render date: 2024-11-28T03:16:42.672Z Has data issue: false hasContentIssue false
  • English
  • Français

Constant slope maps on the extended real line

Published online by Cambridge University Press:  02 May 2017

MICHAŁ MISIUREWICZ  and
SAMUEL ROTH Open the ORCID record for SAMUEL ROTH [Opens in a new window]
MICHAŁ MISIUREWICZ
Affiliation:
Department of Mathematical Sciences, Indiana University-Purdue University Indianapolis, 402 N. Blackford Street, Indianapolis, IN 46202, USA email [email protected], [email protected] Institute of Mathematics, Polish Academy of Sciences, Śniadeckich 8, 00-656 Warsaw, Poland
SAMUEL ROTH
Affiliation:
Department of Mathematical Sciences, Indiana University-Purdue University Indianapolis, 402 N. Blackford Street, Indianapolis, IN 46202, USA email [email protected], [email protected] Institute of Mathematics, Polish Academy of Sciences, Śniadeckich 8, 00-656 Warsaw, Poland Mathematical Institute, Silesian University, Na Rybníčku 1, Opava 746 01, Czech Republic
Get access Rights & Permissions [Opens in a new window]

Abstract

For a transitive countably piecewise monotone Markov interval map we consider the question of whether there exists a conjugate map of constant slope. The answer varies depending on whether the map is continuous or only piecewise continuous, whether it is mixing or not, what slope we consider and whether the conjugate map is defined on a bounded interval, half-line or the whole real line (with the infinities included).

Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 38 , Issue 8 , December 2018 , pp. 3145 - 3169
Copyright
© Cambridge University Press, 2017 

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

Alsedà, Ll., Llibre, J. and Misiurewicz, M.. Combinatorial Dynamics and Entropy in Dimension One (Advanced Series in Nonlinear Dynamics, 5) , 2nd edn. World Scientific, Singapore, 2000.Google Scholar
Bobok, J.. Semiconjugacy to a map of a constant slope. Studia Math. 208 (2012), 213–228.Google Scholar
Bobok, J. and Bruin, H.. Constant slope maps and the Vere-Jones classification. Entropy 18(6) (2016), Paper No. 234, 27 pp.Google Scholar
Bobok, J. and Soukenka, M.. On piecewise affine interval maps with countably many laps. Discrete Contin. Dyn. Syst. 31(3) (2011), 753–762.Google Scholar
Milnor, J. and Thurston, W.. On iterated maps of the interval. Dynamical Systems (Lecture Notes in Mathematics, 1342) . Springer, Berlin, 1988, pp. 465–563.Google Scholar
Misiurewicz, M.. Absolutely continuous measures for certain maps of an interval. Publ. Math. Inst. Hautes Études Sci. 53 (1981), 17–51.Google Scholar
Misiurewicz, M. and Roth, S.. No semiconjugacy to a map of constant slope. Ergod. Th. & Dynam. Sys 36(3) (2016), 875–889.Google Scholar
Parry, W.. Symbolic dynamics and transformations of the unit interval. Trans. Amer. Math. Soc. 122 (1966), 368–378.Google Scholar
Pruitt, W.. Eigenvalues of nonnegative matrices. Ann. Math. Statist. 35 (1964), 1797–1800.Google Scholar
Ruette, S.. On the Vere–Jones classification and existence of maximal measures for countable topological Markov chains. Pacific J. Math. 209(2) (2003), 366–380.Google Scholar
Vere-Jones, D.. Geometric ergodicity in denumerable Markov chains. Quart. J. Math. Oxford Ser. (2) 13 (1962), 7–28.Google Scholar