sure version of the CLT, meaning that ${1 \over
 \log n} \sum_{k=1}^n {1 \over k} \delta_{ {1 \over
    \sqrt{k}} S_k(\psi)(x)}$ converges $\mu {\textrm{-a.e.}}$ to the normal law $\NN(0,{\sigma}^2)$ (see [@LP], [@CG]).\
Let us outline the proof of theorem C (see section \[CC\]). Let ${\omega}: \Sigma \to {\mathbb {P}}^k$ be a coding map provided by theorem A and $\psi \in \UU$. Since ${\omega}$ satisfies $f
\circ {\omega}= {\omega}\circ s$ and ${\omega}_*\nu=\mu$, we are reduced to prove the assertions for $\chi = \psi \circ {\omega}$ on $(\Sigma,s,\nu)$. The points 1 and 2 follow from theorem B(2) and classical arguments. The point 3 is a consequence of theorem B(1) and Philipp-Stout’s theorem ([@PS], section 7). That result relies on an approximation of the partial sums of $(\chi
\circ s^j)_{j \geq 0}$ by a sequence of martingale differences defined with respect to the increasing filtration $(\CC_n)_{n \geq 0}$.

An application to smooth ergodic theory {#jaja}
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Let $\chi_1 \leq \ldots \leq \chi_k$ be the Lyapunov exponents of $\mu$. Briend-Duval [@BrDu1] proved that they are larger than or equal to $\log d^{1/2}$. Since $\mu$ has entropy $\log d^k$, Pesin’s formula $h(\mu) = 2 (\chi_1 + \ldots + \chi_k)$ holds if and only if these exponents 