Noninvertible map

A map is noninvertible when it is not one-to-one and onto on the phase space under consideration. Forward iteration can be well-defined while backward states are absent or nonunique. A zero Jacobian multiplier can provide strong contraction at a fixed point. For the Lozi family, setting $b=0$ sends the whole plane into a line after one iteration, illustrating why a discrete saddle may have a zero stable multiplier without a unique local backward evolution.