are Lipschitz close to the vertical (constant) fibers, and continuous in $z$. This means that $W^s_z$ is the graph of a Lipschitz map $\beta^s_z:F\times M \rightarrow Z$ such that $$\label{eqn:smallcs}
d(\beta^s_z,z)_{C^0}\leq O(\rho), \textrm{ }\textrm{ }\textrm{ }\mLip \beta^s_z \leq O(\rho)$$

$$\label{eqn:holdcs}
d(\beta^s_z ,\beta^s_{z'})_{C^0} \le \frac {d(z,z')^{\a-O(\rho)}}{O(\rho)^\alpha},$$

where $\a=\dsp \frac {\ln \mu}{\ln \mu_-}$.

As was mentioned above, a particularly important case in which the theorem applies is the Smale-Williams solenoid with $Z=S^1$, $F=D$ and $h$ given by .

Fubini revisited {#sub:fubini}
----------------

Let $\Sigma^2_+$ be the set of all sequences $\omega^+ =
\omega_0\omega_1\omega_2 \dots $ of zeroes and ones, infinite to the right with the $(\frac 1 2, \frac 1 2)$ Bernoulli measure. It is known that for almost all such sequences $\omega^+$, any finite word $w$ of any length $n$ is encountered within $\omega^+$ with frequency exactly equal to $2^{-n}$. We are interested in sequences for which this property fails. More precisely, given $\kappa>0$ and $w$ a finite word of length $n$, we say that $\om^+$ is $\kappa,w-$*atypical* if the sequence $$\label{eqn:freq} a_k(\omega^+,w):=\frac {\textrm{number of
occurences of }w\textrm{ among first }k\textrm{ digits of
}\omega^+}{k}$$ has a limit point *outside* $[2^{-n}-\kappa, 2^{-n}+\kappa]$. The sequence $\omega$ defines a point $y=\overline{0,\om^+}\in S^1$ written in 