  We numerically study the two-dimensional, area preserving, web map. When the
map is governed by ergodic behavior, it is, as expected, correctly described by
Boltzmann-Gibbs statistics, based on the additive entropic functional
$S_{BG}[p(x)] = -k\int dx\,p(x) \ln p(x)$. In contrast, possible ergodicity
breakdown and transitory sticky dynamical behavior drag the map into the realm
of generalized $q$-statistics, based on the nonadditive entropic functional
$S_q[p(x)]=k\frac{1-\int dx\,[p(x)]^q}{q-1}$ ($q \in {\cal R}; S_1=S_{BG}$). We
statistically describe the system (probability distribution of the sum of
successive iterates, sensitivity to the initial condition, and entropy
production per unit time) for typical values of the parameter that controls the
ergodicity of the map. For small (large) values of the external parameter $K$,
we observe $q$-Gaussian distributions with $q=1.935\dots$ (Gaussian
distributions), like for the standard map. In contrast, for intermediate values
of $K$, we observe a different scenario, due to the fractal structure of the
trajectories embedded in the chaotic sea. Long-standing non-Gaussian
distributions are characterized in terms of the kurtosis and the box-counting
dimension of chaotic sea.
