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In the late 1970's, P. Coullet and C. Tresser [[#References|[a6]]] and M. Feigenbaum

independently found striking, unexpected features of the transition from simple to chaotic dynamics in one-dimensional dynamical systems (cf. also [[Routes to chaos|Routes to chaos]]). By the example of the family of quadratic mappings  $  f _  \mu  ( x)= 1- \mu x  ^ {2} $
acting (for  $  0 \leq  \mu \leq  2 $) 
on the interval  $  x \in [- 1, 1] $, 
the period-doubling scenario is recalled here. For  $  \mu = 2 $, 
$  f _  \mu  $
has periodic points of every (least) period. Let  $  \mu _ {i }  $
be the infimum of parameter values  $  \mu $
for which  $  f _  \mu  $
has a periodic orbit of least period  $  2 ^ {i } $. 
Then

$$ 
0  <  \mu _ {0}  <  \mu _ {1}  < \dots ,
$$

and

$$ 
\sup  \mu _ {i }   =  \mu _  \infty   \sim  1.401155 \dots .
$$

For  $  \mu _ {i }  < \mu \leq  \mu _ {i+ 1 }  $, 
the dynamics of  $  f _  \mu  $
is described by statements i)–iii) below.

i)  $  f _  \mu  $
has precisely one periodic orbit  $  \Lambda _ {j} $
of (least) period  $  2  ^ {j} $
for each  $  j= 0 \dots i $, 
and no other periodic orbits;

ii) any pair of adjacent points in  $  \Lambda _ {i }  $
is separated by a unique point in  $  \cap _ {j< i }  \Lambda _ {j} $;

iii) with the exception of the (countably many) orbits which land on some  $  \Lambda _ {j} $, 
$  j< i $, 
and stay there, every  $  f _  \mu  $-
orbit tends asymptotically to  $  \Lambda _ {i }  $.

For  $  \mu = \mu _  \infty  $(
when  $  f _  \mu  $
is sometimes called the Feigenbaum mapping), statement i) holds, but with  $  j $
ranging over all non-negative integers, and ii) holds for each  $  i= 0, 1,\dots $; 
furthermore, the following analogue of iii) holds:

iv) (for  $  \mu = \mu _  \infty  $) 
the closure of the orbit of the turning point  $  0 $
is a Cantor set  $  \Lambda _  \infty  $, 
which is the asymptotic limit of every orbit not landing on one of the periodic orbits  $  \Lambda _ {i }  $, 
$  i= 0, 1,\dots $. 
The restricted mapping  $  f _  \mu  \mid  _ {\Lambda _  \infty   } $
is a minimal homeomorphism (the "adding machine for chaos in a dynamical systemadding machine" ).

Finally,  $  \mu = \mu _  \infty  $
is the threshold of "chaos" , in the following sense:

v) for  $  \mu > \mu _  \infty  $, 
$  f _  \mu  $
has infinitely many distinct periodic orbits, and positive topological entropy.

Many features of this "topological" , or combinatorial picture were understood by early researchers in this area, specifically P.J. Myrberg [[#References|[a12]]] and N. Metropolis, M.L. Stein and P.R. Stein [[#References|[a13]]]. They recognized as well that the combinatorial structure of the periodic orbits  $  \Lambda _ {j} $
is rigidly determined by the fact that  $  f _  \mu  $
is unimodal (cf. [[#References|[a14]]]). In essence, the statements above can be formulated for any family of unimodal mappings (cf. ). In fact, the (weak) monotonicity of the  $  \mu _ {i }  $'
s, together with the fact that if  $  \mu < \mu _  \infty  $, 
then  $  f _  \mu  $
must have periodic orbits of least period  $  2  ^ {j} $
for  $  j= 0 \dots i $(
some  $  i $) 
and no others, follows for any family of continuous mappings on the line from Sharkovskii's theorem [[#References|[a16]]], [[#References|[a2]]]; recent work has yielded a more general understanding of the combinatorial structure of periodic orbits for continuous mappings in dimension  $  1 $(
cf. [[#References|[a1]]]).

Coullet, Tresser and Feigenbaum added to the topological picture described above a number of analytic and geometric features:

vi) the convergence  $  \mu _ {i }  \uparrow \mu _  \infty  $
is asymptotically geometric:

$$ 
\lim\limits _ {i \rightarrow \infty } \ 

\frac{\mu _  \infty  - \mu _ {i }  }{\mu _  \infty  - \mu _ {i+ 1 }  }
  =  \delta  \sim  4.669 \dots ;
$$

vii) the periodic orbits scale: let  $  \Lambda _ {i }   ^ {*} $
denote the orbit  $  \Lambda _ {i }  $
for  $  \mu = \mu _ {i+ 1 }  $; 
then

$$ 
\lim\limits _ {i \rightarrow \infty } \ 

\frac{ \mathop{\rm dist} ( 0, \Lambda _ {i }   ^ {*} ) }{ \mathop{\rm dist} ( 0, \Lambda _ {i+ 1 }   ^ {*} ) }
  =  \alpha  \sim \ 
2.5029 \dots .
$$

These statements, formulated for the particular family  $  f _  \mu  $
of quadratic mappings, are technically interesting, but not so striking. However, they observed that v)–vii) hold for a very broad class of unimodal one-parameter families, subject only to trivial "fullness" conditions (essentially that  $  f _ {0} $
has only finitely many periodic orbits while  $  f _ {2} $
has positive entropy) and smoothness (essentially that  $  ( x, \mu ) \rightarrow f _  \mu  ( x) $
is  $  C  ^ {2} $
and each  $  f _  \mu  $
has a non-degenerate critical point). And, sensationally, the constants  $  \delta $
and  $  \alpha $
are independent of the family  $  f _  \mu  $.

In [[#References|[a6]]] and

these assertions were reduced, using ideas from renormalization theory, to certain technical conjectures concerning a doubling operator  $  {\mathcal R} $
acting on an appropriate function space. O. Lanford

(cf. also [[#References|[a3]]], [[#References|[a5]]]) gave a rigorous, computer-assisted proof of the basic conjecture, that  $  {\mathcal R} $
has a saddle-type fixed point with one characteristic multiplier  $  \delta \sim 4.669 \dots $(
the same as in vi)) and stable manifold of codimension  $  1 $. 
D. Sullivan [[#References|[a17]]] showed the uniqueness of this fixed point in the space of "quadratic-like" mappings. The final conjecture, concerning transversality of the stable manifold with certain bifurcation submanifolds, remains unproved. Recently, Sullivan , introducing a number of new ideas, has circumvented this difficulty and provided a rather complete theory of universal features for families of  $  C  ^ {2} $
unimodal mappings. In particular, the asymptotic geometry of the Cantor set  $  \Lambda _  \infty  $(
for  $  \mu = \mu _  \infty  $) 
and of analogous sets appearing at other "threshold" parameter values (the "infinitely renormalizable mappings of bounded type" ) is universal; for example, the set  $  \Lambda _  \infty  $
always has [[Hausdorff dimension|Hausdorff dimension]]  $  0.538045 $. 
Full expositions of this theory are provided in [[#References|[a18]]] and [[#References|[a7]]].

These ideas have been applied as well to circle diffeomorphisms [[#References|[a10]]],

and area-preserving planar diffeomorphisms [[#References|[a4]]], .

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> Ll. Alsedà, J. Llibre, M. Misiurewicz, "Combinatorial dynamics and entropy in one dimension" (to appear)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> L. Block, J. Guckenheimer, M. Misiurewicz, L.-S. Young, "Periodic points and topological entropy of one dimensional maps" Z. Nitecki (ed.) C. Robinson (ed.) , ''Global theory of dynamical systems (Proc. Northwestern Univ., 1979)'' , ''Lect. notes in math.'' , '''819''' , Springer (1980) pp. 18–34 {{MR|0591173}} {{ZBL|0447.58028}} </TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> M. Campanino, H. Epstein, D. Ruelle, "On the existence of Feigenbaum's fixed point" ''Comm. Math. Phys.'' , '''79''' (1981) pp. 261–302 {{MR|612250}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top"> P. Collet, J.-P. Eckmann, H. Koch, "On universality for area-preserving maps of the plane" ''Physica'' , '''3D''' (1981) pp. 457–467 {{MR|0631180}} {{ZBL|1194.37050}} </TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top"> P. Collet, J.-P. Eckmann, O. Lanford, "Universal properties of maps on an interval" ''Comm. Math. Phys.'' , '''76''' (1980) pp. 211–254 {{MR|0588048}} {{ZBL|0455.58024}} </TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top"> P. Coullet, C. Tresser, "Itérations d'endomorphismes et groupe de rénormalisation" ''J. Phys.'' , '''C5''' (1978) pp. 25–28 {{MR|0512110}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top"> W. de Mello, S. van Strien, "One-dimensional dynamics" (to appear)</TD></TR><TR><TD valign="top">[a8a]</TD> <TD valign="top"> M. Feigenbaum, "Quantitative universality for a class of non-linear transformations" ''J. Stat. Phys.'' , '''19''' (1978) pp. 25–52 {{MR|501179}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a8b]</TD> <TD valign="top"> M. Feigenbaum, "The universal metric properties of a non-linear transformation" ''J. Stat. Phys.'' , '''21''' (1979) pp. 669–706 {{MR|555919}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a9a]</TD> <TD valign="top"> L. Jonker, D. Rand, "Bifurcations in one dimension" ''Invent. Math.'' , '''62''' (1981) pp. 347–365 {{MR|0608525}} {{MR|0604832}} {{ZBL|0475.58015}} </TD></TR><TR><TD valign="top">[a9b]</TD> <TD valign="top"> L. Jonker, D. Rand, "Bifurcations in one dimension" ''Invent. Math.'' , '''63''' (1981) pp. 1–16 {{MR|0608525}} {{MR|0604832}} {{ZBL|0475.58015}} </TD></TR><TR><TD valign="top">[a10]</TD> <TD valign="top"> L. Jonker, D. Rand, "Universal properties of maps of the circle with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095650/u09565070.png" />-singularities" ''Comm. Math. Phys.'' , '''90''' (1983) pp. 273–292 {{MR|714439}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a11a]</TD> <TD valign="top"> O. Lanford, "A computer-assisted proof of the Feigenbaum conjectures" ''Bull. Amer. Math. Soc.'' , '''6''' (1982) pp. 427–434 {{MR|0648529}} {{ZBL|0487.58017}} </TD></TR><TR><TD valign="top">[a11b]</TD> <TD valign="top"> O.E. Lanford, "Computer assisted proofs in analysis" A.M. Gleason (ed.) , ''Proc. Internat. Congress Mathematicians (Berkeley, 1986)'' , Amer. Math. Soc. (1987) pp. 1385–1394 {{MR|0934342}} {{ZBL|0676.65039}} </TD></TR><TR><TD valign="top">[a12]</TD> <TD valign="top"> P.J. Myrberg, "Sur l'iteration des polynomes réels quadratiques" ''J. Math. Pures Appl.'' , '''41''' (1962) pp. 339–351 {{MR|0161968}} {{ZBL|0106.04703}} </TD></TR><TR><TD valign="top">[a13]</TD> <TD valign="top"> N. Metropolis, M.L. Stein, P.R. Stein, "On finite limit sets for transformations on the unit interval" ''J. Comb. Theory'' , '''15A''' (1973) pp. 25–44 {{MR|0316636}} {{ZBL|0259.26003}} </TD></TR><TR><TD valign="top">[a14]</TD> <TD valign="top"> W. Thurston, "On iterated maps of the interval" J.C. Alexander (ed.) , ''Dynamical Systems (Proc. Maryland, 1986–7)'' , ''Lect. notes in math.'' , '''1342''' , Springer (1988) pp. 465–563 {{MR|0970571}} {{ZBL|0664.58015}} </TD></TR><TR><TD valign="top">[a15a]</TD> <TD valign="top"> D. Rand, "Universality and renormalization in dynamical systems" T. Bedford (ed.) J. W. Swift (ed.) , ''New directions in dynamical systems'' , Cambridge Univ. Press (1987) pp. 1–56</TD></TR><TR><TD valign="top">[a15b]</TD> <TD valign="top"> D. Rand, "Global phase space universality, smooth conjugacies and renormalisation: the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095650/u09565071.png" /> case." ''Nonlinearity'' , '''1''' (1988) pp. 181–202 {{MR|928952}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a16]</TD> <TD valign="top"> A.N. Sharkovskii, "Coexistence of cycles of a continuous map of the line into itself" ''Ukrain. Mat. Zh.'' , '''16''' (1964) pp. 61–71 (In Russian) {{MR|1415876}} {{MR|1361914}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a17]</TD> <TD valign="top"> D. Sullivan, "Quasiconformal homeomorphisms in dynamics, topology and geometry" A.M. Gleason (ed.) , ''Proc. Internat. Congress Mathematicians (Berkeley, 1986)'' , Amer. Math. Soc. (1987) pp. 1216–1228 {{MR|0934326}} {{ZBL|0698.58030}} </TD></TR><TR><TD valign="top">[a18]</TD> <TD valign="top"> D. Sullivan, "Bounds, quadratic differentials, and renormalization conjectures" , ''Centennial Publ.'' , '''2''' , Amer. Math. Soc. (1991) {{MR|1184622}} {{ZBL|0936.37016}} </TD></TR></table>