Let $f:S^1\to S^1$ be an orientation-preserving circle diffeomorphism with irrational rotation number (see here). Then the system $(S^1,f)$ admits a unique invariant measure, say $\mu_f$.

Let $\displaystyle \lambda(f,x)=\lim_{n\to\infty}\frac{1}{n}\log D_xf^n$ be the Lyapunov exponent at $x$. Then $\lambda(f,x)$ is independent of $x$, and equal to the integral $\int \log D_x f\; d\mu_f(x)$.

If $\log Df$ has bounded variation, then $\lambda(f,x)=0$ (I think it is proved by Herman) and hence $\int \log D_x f\; d\mu_f(x)=0$.

What about the general case? Is $\int \log D_x f\; d\mu_f(x)$ always zero?

Take the Denjoy map for example (a non-transitive diffeomorphism with irrational rotation number).