## Cohomologous functions and the Livsic theorem

Given a dynamical system ${f\colon X\rightarrow X}$ (typically ${X}$ is a compact metric space and ${f}$ is continuous), we commonly encounter real-valued functions ${\varphi\colon X\rightarrow {\mathbb R}}$ in one of the following two roles.

1. An observable function represents a measurement of the system, so that the sequence of functions ${\{\varphi\circ f^n\}}$ represents measurements made at different times, about which we want to make predictions. In the cases we are interested in, these predictions will be probabilistic and will be in terms of the Birkhoff averages ${\frac 1n S_n\varphi(x)}$, where ${S_n\varphi(x) = \sum_{k=0}^{n-1} \varphi(f^k x)}$ is the ${n}$th Birkhoff sum.
2. A potential function is used to assign weights to different trajectories of the system for purposes of selecting an invariant measure with specific dynamical or geometric properties. Generally the orbit segment ${x, f(x), \dots, f^{n-1}(x)}$ is assigned a weight given by ${e^{S_n\varphi(x)}}$. A potential function also assigns weights to invariant measures by integration; in particular, we can study equilibrium measures for ${(X,f,\varphi)}$, which are invariant probability measures maximizing ${h_\mu(f) + \int\varphi\,d\mu}$. (Here ${h_\mu}$ is Kolmogorov–Sinai entropy.)

Before moving on we remark that ${\varphi\colon X\rightarrow[0,\infty)}$ can also play the role of a density function, especially if we are looking for an invariant measure that is absolutely continuous with respect to some reference measure, but for today we will focus on the two roles described above.

Let ${\mathcal{M}_f(X)}$ denote the set of ${f}$-invariant Borel probability measures on ${X}$. Then each ${\varphi \in C(X)}$ induces a map ${\hat\varphi\colon \mathcal{M}_f(X) \rightarrow {\mathbb R}}$ by ${\hat\varphi(\mu) = \int\varphi\,d\mu}$ as in the second item above. It is natural to ask when two functions ${\varphi,\psi \in C(X)}$ have ${\hat\varphi=\hat\psi}$. One immediate observation to make is that the defintion of ${f}$-invariance implies that ${\int\varphi \,d\mu = \int\varphi\circ f \,d\mu}$ for all ${\varphi\in C(X)}$ and ${\mu\in \mathcal{M}_f(X)}$, so that in particular we have ${\hat\varphi = \widehat{\varphi\circ f}}$. Thus the function ${\psi = \varphi - \varphi\circ f}$ has ${\hat\psi = \hat \varphi - \widehat{\varphi\circ f} = 0}$. We call such a function ${\psi}$ a coboundary; we have just shown that

Proposition 1 if ${\psi}$ is a coboundary, then it integrates to ${0}$ with respect to any invariant measure.

Moreover, since integration is linear in ${\varphi}$, we have

Proposition 2 if two functions ${\varphi}$ and ${\psi}$ differ by a coboundary, then ${\hat\varphi=\hat\psi}$, ie., ${\int\varphi\,d\mu = \int\psi\,d\mu}$ for every ${\mu\in \mathcal{M}_f(X)}$.

If ${\varphi}$ and ${\psi}$ differ by a coboundary, we say that they are cohomologous; in this case we can write ${\varphi = \psi + h - h\circ f}$ for some ${h\in C(X)}$, which we call the transfer function. Note that ${\varphi}$ is a coboundary if and only if it is cohomologous to the zero function.

Remark 1 For a discussion of how this is connected to the notion of cohomology in algebraic topology, see Terry Tao’s blog post from December 2008.

It is natural to ask whether cohomology is the only mechanism by which two functions ${\varphi,\psi\in C(X)}$ can have ${\hat\varphi=\hat\psi}$; in other words, do the converses of Propositions 1 and 2 hold?

In general the answer is no, as one can quickly see by letting ${f}$ be an irrational circle rotation, so that the only invariant measure is Lebesgue, and there are many continuous functions on the circle that have Lebesgue integral ${0}$ but are not coboundaries. When ${f}$ has hyperbolic behavior, however, the story is quite different, and this is what we will discuss in this post.

If ${f}$ is a diffeomorphism and ${X}$ is a topologically transitive locally maximal hyperbolic set, then ${f\colon X\rightarrow X}$ satisfies the following result.

Theorem 3 (Closing Lemma) For every ${\epsilon>0}$ there exists ${\delta>0}$ such that if ${x\in X}$ and ${n\geq 0}$ are such that ${d(f^n(x),x) < \delta}$, then there exists ${y\in X}$ such that ${f^n(y) = y}$ and ${d(f^k(y), f^k(x)) < \epsilon}$ for all ${0\leq k < n}$.

Moreover, in this case every Hölder continuous ${\varphi\colon X\rightarrow {\mathbb R}}$ has the following property, introduced by Walters in 1978.

Theorem 4 (Walters Property) For every ${\zeta>0}$ there exist ${\epsilon>0}$ such that if ${x,y\in X}$ and ${n\geq 0}$ are such that ${d(f^k(x),f^k(y)) < \epsilon}$ for all ${0\leq k < n}$, then ${|S_n\varphi(x) - S_n\varphi(y)| < \zeta}$.

Both the Closing Lemma and the Walters Property also hold when ${X}$ is a subshift of finite type and ${\varphi}$ is Hölder continuous.

Using these properties we can formulate and prove an important result.

Theorem 5 (Livsic Theorem) Let ${X}$ be a compact metric space, ${f\colon X\rightarrow X}$ a continuous map satisfying the Closing Lemma and possessing a point whose orbit is dense, and ${\varphi\colon X\rightarrow{\mathbb R}}$ a continuous function satisfying the Walters Property. Then ${\varphi}$ is a coboundary if and only if for every periodic point ${x = f^p(x) \in X}$, we have ${S_p \varphi(x) = 0}$.

The forward implication is immediate: if ${\varphi}$ is a coboundary, then ${\int\varphi\,d\mu = 0}$ for every invariant ${\mu}$, in particular for ${\mu = \frac 1p \sum_{k=0}^{p-1} \delta_{f^k x}}$. Equivalently one can observe that Birkhoff sums of coboundaries have the following behavior: if ${\varphi(x) = h(x) - h(f x)}$, then

$\displaystyle S_n\varphi(x) = \sum_{k=0}^{n-1} \big(h(f^k x) - h(f^{k+1} x)\big) = h(x) - h(f^n x), \ \ \ \ \ (1)$

and consequently ${S_p \varphi(x) = h(x) - h(f^p x) = 0}$ whenever ${x=f^p x}$. We must prove the reverse implication, that if ${S_p\varphi(x)=0}$ whenever ${x=f^p x}$, then ${\varphi}$ is a coboundary. To this end note that if ${h}$ is a continuous (hence uniformly continuous) transfer function for ${\varphi}$, then (1) immediately determines ${h}$ along the entire forward orbit of a point ${y}$ by

$\displaystyle h(f^n y) = h(y) + S_n\varphi(y). \ \ \ \ \ (2)$

(A similar procedure defines ${h}$ along the backward orbit if ${f}$ is invertible.) If ${y}$ is a point whose orbit is dense in ${X}$, then this determines ${h}$ on all of ${X}$ by continuity. Thus it suffices to prove that (2) defines a uniformly continuous function on the orbit of ${y}$. To this end, given ${\zeta>0}$, let ${\epsilon = \epsilon(\zeta)>0}$ be given by the Walters Property, and then let ${\delta = \delta(\epsilon)>0}$ be given by the Closing Lemma. Suppose that ${w,z}$ are two points on the orbit of ${y}$ such that ${d(w,z) < \delta}$. Since ${w,z}$ lie on the same orbit, there is ${n\in {\mathbb N}}$ such that ${f^n(z)=w}$ or ${f^n(w)=z}$. Without loss of generality we assume the first case, so that ${d(z,f^n z) = d(z,w) < \delta}$. By the Closing Lemma there is a periodic point ${y=f^n y}$ such that ${d(f^k y, f^k z) < \epsilon}$ for all ${0\leq k < n}$. By (2) and the Walters Property, this implies that

$\displaystyle |h(w) - h(z)| = |h(f^n z) - h(z)| = |S_n\varphi(z)| \leq |S_n\varphi(x)| + \zeta = \zeta,$

where the last equality uses the fact that Birkhoff sums of ${\varphi}$ vanish around periodic orbits. This proves uniform continuity of ${h}$ on the orbit of ${y}$, so ${h}$ extends uniformly continuously to ${X}$, and it is an easy exercise using continuity of both sides of the equation ${\varphi = h - h\circ f}$ to verify that this relationship continues to hold on all of ${X}$, so that ${\varphi}$ is indeed a coboundary. This completes the proof of the Livsic Theorem.

Let ${X,f}$ be as in the Livsic Theorem; then for any ${\varphi,\psi}$ satisfying the Walters Property, the following are equivalent:

1. ${\varphi = \psi + h - h\circ f}$ for some ${h\in C(X)}$;
2. ${\int\varphi\,d\mu = \int\psi\,d\mu}$ for all ${\mu\in\mathcal{M}_f(X)}$;
3. ${\int\varphi\,d\mu = \int\psi\,d\mu}$ for all periodic orbit measures.

Here is one final remark on a specific example of cohomologous potentials: one immediately sees that ${\int \frac 1nS_n\varphi \,d\mu = \int \varphi\,d\mu}$ for every ${\mu\in\mathcal{M}_f(X)}$, and so under the conditions of the Livsic Theorem one can conclude that each Birkhoff average ${\frac 1nS_n\varphi}$ is cohomologous to ${\varphi}$. In fact this can be shown directly, without using the Livsic Theorem, by putting

\displaystyle \begin{aligned} h(x) &= \frac 1n\big((n-1) \varphi(x) + (n-2)\varphi(f x) + (n-3)\varphi(f^2 x) + \cdots + \varphi(f^{n-2} x)\big) \\ &= \frac 1n\sum_{k=0}^{n-1} (n-1-k) \varphi(f^{k-1} x). \end{aligned}

Then we have

\displaystyle \begin{aligned} h(x) - h(fx) &= \frac 1n\big((n-1) \varphi(x) + (n-2)\varphi(f x) + \cdots + \varphi(f^{n-2} x)\big) \\ &\quad\quad - \frac 1n\big((n-1) \varphi(f x) + (n-2)\varphi(f^2 x) + \cdots + \varphi(f^{n-1} x)\big) \\ &= \frac{n-1}n \varphi(x) - \frac 1n\big( \varphi(fx) + \varphi(f^2x) + \cdots + \varphi(f^{n-1} x) \big) \\ &= \varphi(x) - \frac 1n S_n\varphi(x), \end{aligned}

which demonstrates that ${\varphi}$ and ${\frac 1nS_n\varphi}$ are cohomologous.