  The main purpose is to show that Feigenbaum delta constant is much more
universal than believed. The paper is mainly devoted to period-doubling
processes in families in one parameter of endomorphisms of the interval and
consider generalizations of the Feigenbaum delta constant. We formulate the
so-called parenthesis permeability hypothesis, a conjecture that holds for all
types of bifurcation (i.e. for flip, fold, pitchfork and transcritical
bifurcations, which states that under some conditions two or three different
functions may have exactly the same bifurcation points.
  We propose a conjecture that considerably relaxes David Singer conditions for
endomorphism families to generate at most one stable orbit, showing that
Feigenbaum constant appears also in some classes of functions that have more
than one maximum and have positive Schwarzian in at least one sub-interval.
This version contains more arguments in favor of an even greater Feiganbaum
constant universality.
