Abstract
The Abramov-Rokhlin formula states that the entropy of a measure-preserving transformation S equals the sum of the entropy of a factor T of S and the entropy of S relative to T. We prove this formula for non-invertible transformations and apply it to skew-product transformations.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
L. M. Abramov and V. A. Rokhlin, The entropy of a skew product of measurepreserving transformations, Amer. Math. Soc. Transl. Ser. 2, 48 (1966), 255–265
T. Bogenschütz, Entropy for random dynamical systems, Report No. 235, Institut für Dynamische Systeme, Universität Bremen 1990
H. Crauel, Lyapunov exponents and invariant measures of stochastic systems on manifolds, in: L. Arnold and V. Wihstutz (eds.), Lyapunov Exponents, Proceedings, Bremen 1984, Lecture Notes in Mathematics 1186, Springer 1986
P. Hulse, Sequence entropy relative to an invariant σ-algebra, J. London Math. Soc. (2) 33 (1986), 59–72
Y. Kifer, Ergodic Theory of Random Transformations, Birkhäuser 1986
B. Kamiński, An axiomatic definition of the entropy of a ℤd-action on a Lebesgue space, Studia Mathematica, T. XCVI (1990), 135–144
F. Ledrappier and P. Walters, A relativised variational principle for continuous transformations, J. London Math. Soc. (2) 16 (1977), 568–576
K. Petersen, Ergodic Theory, Cambridge University Press 1983
J.-P. Thouvenot, Quelques propriétés des systèmes dynamiques qui se décomposent en un produit de deux systèmes dont l’un est un schéma de Bernoulli, Israel J. Math. 21 (1975), 177–207
P. Walters, Relative pressure, relative equilibrium states, compensation functions and many-to-one codes between subshifts, Trans. Amer. Math. Soc. 296 (1986), 1–31
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 1992 Springer-Verlag
About this paper
Cite this paper
Bogenschütz, T., Crauel, H. (1992). The Abramov-Rokhlin formula. In: Krengel, U., Richter, K., Warstat, V. (eds) Ergodic Theory and Related Topics III. Lecture Notes in Mathematics, vol 1514. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0097526
Download citation
DOI: https://doi.org/10.1007/BFb0097526
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-55444-8
Online ISBN: 978-3-540-47076-2
eBook Packages: Springer Book Archive