Intuition of cocycles and their use in dynamical systems

I’ve come across several papers and lectures that use co-cycles to talk about dynamics on a manifold. However, I haven’t come across an actual definition of what a co-cycle is. Could someone give a brief description for intuition and provide reference material for deeper study? If possible, I’d like to take a dynamical viewpoint on this concept. Thanks!

In addition to the comment above, I would also like to give a reference to cocycles. This is from "Handbook of Dynamical Systems by B. Hasselblatt, A. Katok"

k. Cocycles. A central role in many aspects of dynamical systems is played by cocycles. A 1-cocycle with values in a topological group $H$ over an action $\Phi : G \times X \to X$ is defined to be a map $\alpha : G \times X \to H$, continuous in $G$, such that

$$\alpha(g_1 g_2, x) = \alpha \big ( g_2, \Phi^{g_1}(x) \big ) \alpha(g_1,x).$$

Two cocycles $\alpha, \beta$ are said to be cohomologous if there is a map $C : X \to H$, called a transfer function such that

$$\alpha(g,x) = C \big ( \Phi^g(x) \big ) \beta(g,x) C(x)^{-1}.$$

A cocycle is said to be a coboundary if it is cohomologous to the identity in $H$.

The notion of regularity of a cocycle as a function on the phase space depends on the structure of the phase space (measurable, topological, smooth). Sometimes it turns out to be natural to consider cohomology of cocycles in a sense weaker than the ambient structure. I.e., the transfer function may only need to be of some lower regularity than the cocycles themselves.

Note that a cocycle independent of $x$ is given by a homomorphism $G \to H$. If $H$ is Abelian then one can define a product of cocycles, coboundaries form a subgroup of the Abelian group of all cocycles, and hence the set of cohomology classes has a group structure. Formally this is the first cohomology group of $G$ acting on $X$ with coefficients in $H$. In dynamics the regularity of the cocycles and transfer functions plays a central role and in the presence of nontrivial asymptotic behavior the calculation of the cohomology groups only rarely reduces to formal algebraic manipulations. Higher cohomology groups can be defined following the general prescription of homological algebra [82].

If $H$ is non-Abelian the set of cohomology classes does not possess any group structure. Depending on the structure of the space on which the dynamics is defined, there are cocycles naturally associated with the dynamics, such as the Radon-Nikodym cocycle for transformations with quasi-invariant measures (the Jacobian cocycle in the case of smooth dynamics. Section 5.2k).

Hope this helps!