Homology of a dynamical system
cohomology of a dynamical system
One of the invariants in ergodic theory, the construction of which recalls the construction of the cohomology of a group [1]. In the simplest case of the one-dimensional (co)homology group of the cascade obtained by iterating an automorphism
of a measure space
, the definition is equivalent to the following one. Let
be the additive group of all measurable functions on
(or, respectively, the multiplicative group of all measurable functions
for which
almost-everywhere). The additive (multiplicative) (co) boundary of a function
is the function
(or, respectively,
). If one denotes the set of all (co)boundaries by
, then one may define the additive (or, respectively, multiplicative) (co)homology group
as the quotient group
. Narrower classes of functions rather than all measurable functions may also be considered. The homology groups of a dynamical system are invariants of a trajectory isomorphism (for details on
see [2]).
The homology of a dynamical system has not yet (1977) been computed for even a single non-trivial example. The use of "homological" concepts in ergodic theory stems from the fact that in many real cases it may be important to know (and it is sometimes actually known) whether some given function is or is not a coboundary.
References
[1] | A.A. Kirillov, "Dynamical systems, factors and representations of groups" Russian Math. Surveys , 22 : 5 (1967) pp. 63–75 Uspekhi Mat. Nauk , 22 : 5 (1967) pp. 67–80 |
[2] | A.M. Stepin, "Cohomologies of automorphism groups of a Lebesgue space" Funct. Anal. Appl. , 5 (1971) pp. 167–168 Funktsional. Anal. i Prilozhen. , 5 : 2 (1971) pp. 91–92 |
Comments
For related results on cocycles, see e.g. [a1] and the references given there (e.g., to work of G.W. Mackey).
The study of cohomology for abstract (minimal) topological dynamical systems (i.e., dynamical systems consisting of arbitrary topological groups acting on a compact space) was initiated in [a3]. For further developments, see [a2].
References
[a1] | V.Ya. Golodets, S.D. Sinelshchikov, "Locally compact groups appearing as ranges of cocycles of ergodic ![]() |
[a2] | R. Ellis, "Cohomology of groups and almost periodic extensions of minimal sets" Ergodic Theory and Dynamical Systems , 1 (1981) pp. 49–64 |
[a3] | K.E. Petersen, "Extension of minimal transformation groups" Math. Systems Theory , 5 (1971) pp. 365–375 |
Homology of a dynamical system. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Homology_of_a_dynamical_system&oldid=13252