(6,6)(0,0) (0.5,0.53) (0.2,0.5) (0.45,0.25) (0.25,5.5) (5.5,0.25) (0.5,0.5)[(5,5)]{} (0.5,5.5)[(0,1)[0.25]{}]{} (5.5,0.5)[(1,0)[0.25]{}]{} (0.45,5.85) (5.8,0.45)

\[fig:fig1\]

The maximal invariant set $X^+$ for the map (\[eqeg1\]) is shown in Fig. 2. The results in Theorem \[thmposmeas\] imply that $$\frac{1}{4}\leq \ell(X^+)\leq \frac{3}{4}$$ but this is clearly not very precise.

We have obtained a numerical estimate by iterating a grid of initial points distributed uniformly on the square, dividing the unit square into boxes and counting the occupied cells. To avoid the transient effects some number (say, the first hundred) of images are not marked on the graph. Applying this method we obtained $V\sim 0.287$, however the precision of this result is limited by a highly complicated structure of some fragments of the invariant set and a very slow convergence of transients.

(6,6)(0,0) (0.5,0.53) (0.2,0.5) (0.45,0.25) (0.25,5.5) (5.5,0.25) (0.5,0.5)[(5,5)]{} (0.5,5.5)[(0,1)[0.25]{}]{} (5.5,0.5)[(1,0)[0.25]{}]{} (0.45,5.85) (5.8,0.45)

\[fig:bif\]

By using the structure noted in Section \[secmaximal\] we show numerical approximations of $X_B^+$ as $B$ changes in the map (\[pmap\]) are shown in Figure. \[fig:bif\] and notice that the bifurcation diagram of $X_B^+$ has a regular structure $$\begin{array}{cc}
{1 \over 2} < B < {5 \over 6} & \ell(X^+_B) = {1 \over 2}\\
{5 \over 6} < B < {7 \over 8} 