  We investigate a generalisation of the logistic map as $
x_{n+1}=1-ax_{n}\otimes_{q_{map}} x_{n}$ ($-1 \le x_{n} \le 1$, $0<a\le2$)
where $\otimes_q$ stands for a generalisation of the ordinary product, known as
$q$-product [Borges, E.P. Physica A {\bf 340}, 95 (2004)]. The usual product,
and consequently the usual logistic map, is recovered in the limit $q\to 1$,
The tent map is also a particular case for $q_{map}\to\infty$. The
generalisation of this (and others) algebraic operator has been widely used
within nonextensive statistical mechanics context (see C. Tsallis, {\em
Introduction to Nonextensive Statistical Mechanics}, Springer, NY, 2009). We
focus the analysis for $q_{map}>1$ at the edge of chaos, particularly at the
first critical point $a_c$, that depends on the value of $q_{map}$. Bifurcation
diagrams, sensitivity to initial conditions, fractal dimension and rate of
entropy growth are evaluated at $a_c(q_{map})$, and connections with
nonextensive statistical mechanics are explored.
