  A generalized method is proposed to compose new orbits from a given chaotic
map. The method provides an approach to examine discrete-time chaotic maps in a
"deep-zoom" manner by using $k$-digits to the right from the decimal separator
of a given point from the underlying chaotic map. Interesting phenomena have
been identified. Rapid randomization was observed, i.e. chaotic patterns tend
to become indistinguishable, when compared to the original orbits of the
underlying chaotic map. Our results were presented using different graphical
analyses (i.e., time-evolution, bifurcation diagram, Lyapunov exponent,
Poincar\'e diagram and frequency distribution). Moreover, taking advantage of
this randomization improvement, we propose a Pseudo-Random Number Generator
(PRNG) based on the $k$-logistic map. The pseudo-random qualities of the
proposed PRNG passed both tests successfully, i.e. DIEHARD and NIST, and were
comparable with other traditional PRNGs such as the Mersenne Twister. The
results suggest that simple maps such as the logistic map can be considered as
good PRNG methods.
