Noninvertible Maps and Critical Sets - Nonlinear Oligopolies: Stability and Bifurcations - page 287

Appendix C
Noninvertible Maps and Critical Sets

In this appendix we give some definitions, properties and simple examples of
discrete dynamical systems represented by the iteration of noninvertible maps.

C.1 Definitions and Simple Examples

n ,definedby x 0 D T. x /, transforms a point x 2 S into a
unique point x 0 2 S. The point x 0 is called the rank-1 image of x , and a point x such
that T. x / D x 0 is a rank -1 preimage of x 0 .
If x ¤ y implies T. x / ¤ T. y / for each x , y in S,thenT is an invertible map
in S, because the inverse mapping x D T 1 . x 0 / is uniquely defined. Otherwise T
is said to be a noninvertible map , because points x exist that have several rank-1
preimages, i.e., the inverse relation x D T 1 . x 0 / is multivalued. So, noninvertible
means “many-to-one”, that is, distinct points x

AmapT W S ! S, S R

¤

y may have the same image,

T. x / D T. y / D x 0 .
Geometrically, the action of a noninvertible map can be thought of as “folding
and pleating” the space S, so that distinct points are mapped into the same point.
This is equivalently stated by saying that several inverses are defined in some points
of S, and these inverses “unfold” S.
For a noninvertible map, S can be subdivided into regions Z k , k 0, whose
points have k distinct rank-1 preimages. Generally, for a continuous map, as the
point x 0 varies in

n , pairs of preimages appear or disappear as this point crosses the
boundaries separating different regions. Hence, such boundaries are characterized
by the presence of at least two coincident (merging) preimages. This leads us to
the definition of the critical sets , one of the distinguishing features of noninvertible
maps (see Gumowski and Mira (1980), Mira et al. (1996)):

R

Definition C.1. The critical set CS of a continuous map T is defined as the locus
of points having at least two coincident rank 1 preimages, located on a set CS 1 ,
called the set of merging preimages .

The critical set CS is generally formed by .n 1/-dimensional hypersurfaces

n , and portions of CS separate regions Z k of the phase space characterized

of R

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