Intuition of cocycles and their use in dynamical systems

Question

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!

Answer 1

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!

Answer 2

Regarding the intuition for cocycles in dynamics, I think a good example is the derivative of a map. Consider a continuously differentiable self-map $f:M\to M$ of a differentiable manifold $M$, say of dimension $d$. Iterates of $f$

$$f, f\circ f, f\circ f\circ f,..., f^{ n}=f\circ f\circ \cdots\circ f,...$$

constitute a smooth dynamical system on $M$. If $x\in M$ is a point, then we have its orbit:

$$x\mapsto f(x)\mapsto f^{ 2}(x)\mapsto f^{ 3}(x)\mapsto \cdots \mapsto f^{ n}(x)\mapsto\cdots$$

Since $f$ is differentiable, it has a well defined derivative $Tf: TM\to TM$ that takes any tangent vector $v\in T_xM$ to $M$ at $x$ to some tangent vector $T_xf(v)\in T_{f(x)}M$ at $f(x)$. Likewise, any iterate $f^n$ is also differentiable, and

$$T_x(f^n): T_x M\to T_{f^n(x)}M.$$

So the derivatives of iterates of $f^n$ at $x$ follow the orbit of $x$. How are the derivatives of iterates related to the iterates of derivatives? Of course, by the chain rule we have

$$T_x(f^2)= T_{f(x)}f \,\circ\, T_x f,\,\, T_x(f^3) = T_{f^2(x)}f\,\circ\, T_{f(x)}f\,\circ\, T_x f$$

and in general

$$T_x(f^n)= T_{f^{n-1}(x)}f \,\circ\, T_{f^{n-2}(x)}f\,\circ \, \cdots T_{f(x)}f\,\circ \, T_xf. $$

Even more generally we have

$$T_x(f^{n+m})=T_{f^m(x)}(f^n)\,\circ\, T_x(f^m). $$

We can abbreviate this last equation by defining $\mathcal{T}(n,x)= T_x(f^n)$. Indeed, with this new notation we have

$$\mathcal{T}(n+m,x)= \mathcal{T}(n,f^m(x))\,\circ \mathcal{T}(m,x).$$

Let us fix an auxilliary continuous Riemannian metric on $TM$ so that we can identify each $T_x M$ with $\mathbb{R}^d$ and each derivative with a $d\times d$ matrix with real entries. Then we have that

$$\mathcal{T}:\mathbb{Z}_{\geq0}\times M\to \operatorname{Mat}(d,\mathbb{R}).$$

(Alternatively one can use the fact that any vector bundle over a manifold is measure theoretically trivial.)

This is called the derivative cocycle over $f$; it allows one to extend the dynamics on the points of $M$ to dynamics on the points and velocities of $M$, i.e. it allows one to extend dynamics on the configuration space to the phase space. One of its uses in dynamics is to quantify the speed at which orbits of nearby points separate from each other asymptotically (by way of Lyapunov exponents). In general one common use of cocycles in dynamics is to extend group action to larger spaces.

---

Another standard example would be as follows: now consider a continuous self-map $f:M\to M$ of a compact metric space $M$. Let us fix a continuous observable function $\Phi: M\to \mathbb{R}$. If $x\in M$ is a point, we have again the orbit of $x$ as above. Let us tally all numbers the observable $\Phi$ outputs along the orbit of $x$ and add them up:

\begin{align*} &\Phi(x),\Phi\circ f(x),\Phi\circ f^2(x),...,\Phi\circ f^{n-1}(x),... \\ &\text{ gives } \Phi(x),\Phi(x)+\Phi\circ f(x),...,\Phi(x)+\Phi\circ f(x)+\Phi\circ f^2(x)+\cdots+\Phi\circ f^{n-1}(x),... \end{align*}

How do the sums of iterates relate to the iterates of sums? Putting $\mathcal{S}(n,\Phi)(x)=\sum_{k=0}^{n-1}\Phi\circ f^{k}(x)$ and writing out the sums explicitly we again have

$$\mathcal{S}(n+m,\Phi)(x) = \mathcal{S}(n,\Phi\circ f^m)(x)+\mathcal{S}(m,\Phi)(x).$$

Thus we have $\mathcal{S}:\mathbb{Z}_{\geq0}\times C^0(M;\mathbb{R})\to C^0(M;\mathbb{R})$. More succinctly, let us put $U(\Phi)=\Phi\circ f$. Then we have

$$\mathcal{S}(n+m,\Phi)=\mathcal{S}(n,U^m(\Phi))+\mathcal{S}(m,\Phi).$$

This is called the Birkhoff cocycle over the action of $f$ (or the Koopman operator $U$ of $f$) on the space of continuous observables on $M$ (or more typically simply Birkhoff sums of $f$). The Birkhoff cocycle (in various spaces of observables) is fundamental in ergodic theory, as the ergodic theorem is about the existence and further properties of asymptotic averages

$$\lim_{n\to \infty} \dfrac{1}{n}\mathcal{S}(n,\Phi)(x).$$

To compare, note that the operation we used in the target of $\mathcal{T}$ was matrix multiplication/composition, whereas in the target of $\mathcal{S}$ we used the operation of pointwise addition.

(As an exercise, it might be worthwhile to think about which action $\mathcal{S}$ allows one to extend to which action.)

(A slight variant of $\mathcal{S}$ can be used to construct a special flow over a measure-preserving automorphism, see e.g. https://math.stackexchange.com/a/4341581/169085 .)

---

There are (at least) two other standard starting points for cocycles in dynamical systems; one is time changes (e.g. as in the topological classification of flows) and the other is random dynamics (e.g. consider multiplying two matrices $A$ and $B$ infinitely many times, where the order of multiplication is determined by a coin flip); these are, in my opinion, more specialized perspectives, so I won't get into them here.