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In the late 1970's, P. Coullet and C. Tresser [[#References|[a6]]] and M. Feigenbaum
 
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095650/u0956501.png" /> acting (for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095650/u0956502.png" />) on the interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095650/u0956503.png" />, the period-doubling scenario is recalled here. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095650/u0956504.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095650/u0956505.png" /> has periodic points of every (least) period. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095650/u0956506.png" /> be the infimum of parameter values <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095650/u0956507.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095650/u0956508.png" /> has a periodic orbit of least period <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095650/u0956509.png" />. Then
+
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095650/u09565010.png" /></td> </tr></table>
+
$$
 +
< \mu _ {0< \mu _ {1}  < \dots ,
 +
$$
  
 
and
 
and
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095650/u09565011.png" /></td> </tr></table>
+
$$
 +
\sup  \mu _ {i }  = \mu _  \infty  \sim  1.401155 \dots .
 +
$$
  
For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095650/u09565012.png" />, the dynamics of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095650/u09565013.png" /> is described by statements i)–iii) below.
+
For $  \mu _ {i }  < \mu \leq  \mu _ {i+ 1 }  $,  
 +
the dynamics of $  f _  \mu  $
 +
is described by statements i)–iii) below.
  
i) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095650/u09565014.png" /> has precisely one periodic orbit <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095650/u09565015.png" /> of (least) period <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095650/u09565016.png" /> for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095650/u09565017.png" />, and no other periodic orbits;
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095650/u09565018.png" /> is separated by a unique point in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095650/u09565019.png" />;
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095650/u09565020.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095650/u09565021.png" />, and stay there, every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095650/u09565022.png" />-orbit tends asymptotically to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095650/u09565023.png" />.
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095650/u09565024.png" /> (when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095650/u09565025.png" /> is sometimes called the Feigenbaum mapping), statement i) holds, but with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095650/u09565026.png" /> ranging over all non-negative integers, and ii) holds for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095650/u09565027.png" />; furthermore, the following analogue of iii) holds:
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095650/u09565028.png" />) the closure of the orbit of the turning point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095650/u09565029.png" /> is a Cantor set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095650/u09565030.png" />, which is the asymptotic limit of every orbit not landing on one of the periodic orbits <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095650/u09565031.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095650/u09565032.png" />. The restricted mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095650/u09565033.png" /> is a minimal homeomorphism (the "adding machine for chaos in a dynamical systemadding machine" ).
+
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, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095650/u09565034.png" /> is the threshold of "chaos" , in the following sense:
+
Finally, $  \mu = \mu _  \infty  $
 +
is the threshold of "chaos" , in the following sense:
  
v) for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095650/u09565035.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095650/u09565036.png" /> has infinitely many distinct periodic orbits, and positive topological entropy.
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095650/u09565037.png" /> is rigidly determined by the fact that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095650/u09565038.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095650/u09565039.png" />'s, together with the fact that if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095650/u09565040.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095650/u09565041.png" /> must have periodic orbits of least period <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095650/u09565042.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095650/u09565043.png" /> (some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095650/u09565044.png" />) 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095650/u09565045.png" /> (cf. [[#References|[a1]]]).
+
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:
 
Coullet, Tresser and Feigenbaum added to the topological picture described above a number of analytic and geometric features:
  
vi) the convergence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095650/u09565046.png" /> is asymptotically geometric:
+
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 ;
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095650/u09565047.png" /></td> </tr></table>
+
vii) the periodic orbits scale: let  $  \Lambda _ {i }  ^ {*} $
 +
denote the orbit  $  \Lambda _ {i }  $
 +
for  $  \mu = \mu _ {i+ 1 }  $;  
 +
then
  
vii) the periodic orbits scale: let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095650/u09565048.png" /> denote the orbit <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095650/u09565049.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095650/u09565050.png" />; then
+
$$
 +
\lim\limits _ {i \rightarrow \infty } \
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095650/u09565051.png" /></td> </tr></table>
+
\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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095650/u09565052.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095650/u09565053.png" /> has only finitely many periodic orbits while <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095650/u09565054.png" /> has positive entropy) and smoothness (essentially that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095650/u09565055.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095650/u09565056.png" /> and each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095650/u09565057.png" /> has a non-degenerate critical point). And, sensationally, the constants <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095650/u09565058.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095650/u09565059.png" /> are independent of the family <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095650/u09565060.png" />.
+
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
 
In [[#References|[a6]]] and
  
these assertions were reduced, using ideas from renormalization theory, to certain technical conjectures concerning a doubling operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095650/u09565061.png" /> acting on an appropriate function space. O. Lanford
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095650/u09565062.png" /> has a saddle-type fixed point with one characteristic multiplier <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095650/u09565063.png" /> (the same as in vi)) and stable manifold of codimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095650/u09565064.png" />. 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095650/u09565065.png" /> unimodal mappings. In particular, the asymptotic geometry of the Cantor set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095650/u09565066.png" /> (for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095650/u09565067.png" />) and of analogous sets appearing at other "threshold" parameter values (the "infinitely renormalizable mappings of bounded type" ) is universal; for example, the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095650/u09565068.png" /> always has [[Hausdorff dimension|Hausdorff dimension]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095650/u09565069.png" />. Full expositions of this theory are provided in [[#References|[a18]]] and [[#References|[a7]]].
+
(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]]],
 
These ideas have been applied as well to circle diffeomorphisms [[#References|[a10]]],

Latest revision as of 08:27, 6 June 2020


In the late 1970's, P. Coullet and C. Tresser [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). 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 [a12] and N. Metropolis, M.L. Stein and P.R. Stein [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. [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 [a16], [a2]; recent work has yielded a more general understanding of the combinatorial structure of periodic orbits for continuous mappings in dimension $ 1 $( cf. [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 [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 [a3], [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 [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 $ 0.538045 $. Full expositions of this theory are provided in [a18] and [a7].

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

and area-preserving planar diffeomorphisms [a4], .

References

[a1] Ll. Alsedà, J. Llibre, M. Misiurewicz, "Combinatorial dynamics and entropy in one dimension" (to appear)
[a2] 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 MR0591173 Zbl 0447.58028
[a3] M. Campanino, H. Epstein, D. Ruelle, "On the existence of Feigenbaum's fixed point" Comm. Math. Phys. , 79 (1981) pp. 261–302 MR612250
[a4] P. Collet, J.-P. Eckmann, H. Koch, "On universality for area-preserving maps of the plane" Physica , 3D (1981) pp. 457–467 MR0631180 Zbl 1194.37050
[a5] P. Collet, J.-P. Eckmann, O. Lanford, "Universal properties of maps on an interval" Comm. Math. Phys. , 76 (1980) pp. 211–254 MR0588048 Zbl 0455.58024
[a6] P. Coullet, C. Tresser, "Itérations d'endomorphismes et groupe de rénormalisation" J. Phys. , C5 (1978) pp. 25–28 MR0512110
[a7] W. de Mello, S. van Strien, "One-dimensional dynamics" (to appear)
[a8a] M. Feigenbaum, "Quantitative universality for a class of non-linear transformations" J. Stat. Phys. , 19 (1978) pp. 25–52 MR501179
[a8b] M. Feigenbaum, "The universal metric properties of a non-linear transformation" J. Stat. Phys. , 21 (1979) pp. 669–706 MR555919
[a9a] L. Jonker, D. Rand, "Bifurcations in one dimension" Invent. Math. , 62 (1981) pp. 347–365 MR0608525 MR0604832 Zbl 0475.58015
[a9b] L. Jonker, D. Rand, "Bifurcations in one dimension" Invent. Math. , 63 (1981) pp. 1–16 MR0608525 MR0604832 Zbl 0475.58015
[a10] L. Jonker, D. Rand, "Universal properties of maps of the circle with -singularities" Comm. Math. Phys. , 90 (1983) pp. 273–292 MR714439
[a11a] O. Lanford, "A computer-assisted proof of the Feigenbaum conjectures" Bull. Amer. Math. Soc. , 6 (1982) pp. 427–434 MR0648529 Zbl 0487.58017
[a11b] 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 MR0934342 Zbl 0676.65039
[a12] P.J. Myrberg, "Sur l'iteration des polynomes réels quadratiques" J. Math. Pures Appl. , 41 (1962) pp. 339–351 MR0161968 Zbl 0106.04703
[a13] 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 MR0316636 Zbl 0259.26003
[a14] 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 MR0970571 Zbl 0664.58015
[a15a] 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
[a15b] D. Rand, "Global phase space universality, smooth conjugacies and renormalisation: the case." Nonlinearity , 1 (1988) pp. 181–202 MR928952
[a16] 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) MR1415876 MR1361914
[a17] 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 MR0934326 Zbl 0698.58030
[a18] D. Sullivan, "Bounds, quadratic differentials, and renormalization conjectures" , Centennial Publ. , 2 , Amer. Math. Soc. (1991) MR1184622 Zbl 0936.37016
How to Cite This Entry:
Universal behaviour in dynamical systems. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Universal_behaviour_in_dynamical_systems&oldid=49089
This article was adapted from an original article by Z. Nitecki (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article