View source for Homology of a dynamical system

''cohomology of a dynamical system''

One of the invariants in [[Ergodic theory|ergodic theory]], the construction of which recalls the construction of the cohomology of a group [[#References|[1]]]. In the simplest case of the one-dimensional (co)homology group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047820/h0478201.png" /> of the cascade obtained by iterating an automorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047820/h0478202.png" /> of a measure space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047820/h0478203.png" />, the definition is equivalent to the following one. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047820/h0478204.png" /> be the additive group of all measurable functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047820/h0478205.png" /> (or, respectively, the multiplicative group of all measurable functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047820/h0478206.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047820/h0478207.png" /> almost-everywhere). The additive (multiplicative) (co) boundary of a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047820/h0478208.png" /> is the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047820/h0478209.png" /> (or, respectively, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047820/h04782010.png" />). If one denotes the set of all (co)boundaries by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047820/h04782011.png" />, then one may define the additive (or, respectively, multiplicative) (co)homology group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047820/h04782012.png" /> as the quotient group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047820/h04782013.png" />. 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047820/h04782014.png" /> see [[#References|[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====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  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</TD></TR></table>



====Comments====
For related results on cocycles, see e.g. [[#References|[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 [[#References|[a3]]]. For further developments, see [[#References|[a2]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  V.Ya. Golodets,   S.D. Sinelshchikov,   "Locally compact groups appearing as ranges of cocycles of ergodic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047820/h04782015.png" />-actions"  ''Ergodic Theory and Dynamical Systems'' , '''5'''  (1985)  pp. 47–57</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  R. Ellis,   "Cohomology of groups and almost periodic extensions of minimal sets"  ''Ergodic Theory and Dynamical Systems'' , '''1'''  (1981)  pp. 49–64</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  K.E. Petersen,   "Extension of minimal transformation groups"  ''Math. Systems Theory'' , '''5'''  (1971)  pp. 365–375</TD></TR></table>