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 is denoted by $\lambda(f)$.

**Question.** Is $\lambda(f)$ always zero?

Note that $\lambda(f)$ equals to the integral $\int \log D_x f\; d\mu_f(x)$. Take the Denjoy map for example (a non-transitive diffeomorphism with irrational rotation number).

**Remark.** If $\log Df$ has bounded variation, then Denjoy proved that $e^{-V}\le Df^{q_n}(x)\le e^V$ for all $x$ (a *much stronger* property), where $q_n, n\ge 1$ are the denominators of rational approximates. In particular this implies $\lambda(f,x)=0$ for all $x$.

**Edit again.** I thought the integral might be easier to compute. Then I realized that the estimation of $\lambda(f)$ is quite straight forward, and answered my own question.