View source for Topological entropy

<!--
t0930401.png
$#A+1 = 53 n = 0
$#C+1 = 53 : ~/encyclopedia/old_files/data/T093/T.0903040 Topological entropy
Automatically converted into TeX, above some diagnostics.
Please remove this comment and the {{TEX|auto}} line below,
if TeX found to be correct.
-->

{{TEX|auto}}
{{TEX|done}}

A concept in [[Topological dynamics|topological dynamics]] and [[Ergodic theory|ergodic theory]] analogous to the [[Metric entropy|metric entropy]] of dynamical systems (introduced in [[#References|[1]]]). For an open covering  $  \mathfrak A $
of a compactum  $  X $, 
let  $  H ( \mathfrak A ) $
denote the logarithm (usually to base 2) of the smallest number of elements of the covering that can cover  $  X $. 
If  $  S:  X \rightarrow X $
is a continuous mapping, then the limit

$$ 
h ( S, \mathfrak A )  = \ 
\lim\limits _ {n \rightarrow \infty } \ 
{
\frac{1}{n}
 } H
( \mathfrak A \lor S  ^ {-1}
\mathfrak A \lor \dots \lor
S ^ {- n + 1 } \mathfrak A )
$$

exists, where  $  \mathfrak A \lor \mathfrak B $
is the covering whose elements are all the non-empty intersections of the elements of  $  \mathfrak A $
and  $  \mathfrak B $. 
The topological entropy  $  h _ { \mathop{\rm top}  } ( S) $
is defined to be the supremum of  $  h ( S, \mathfrak A ) $
over all possible  $  \mathfrak A $. 
There is an equivalent definition in the metric case: For a metric  $  \rho $, 
let  $  K _  \epsilon  ( X, \rho ) $
denote the largest number of points of  $  X $
with pairwise distance greater than  $  \epsilon $. 
Then

$$ 
h _ { \mathop{\rm top}  } ( S)  = \ 
\lim\limits _ {\epsilon \rightarrow 0 } \ 
\lim\limits _ {n \rightarrow \infty } \ 
{
\frac{1}{n}
 }   \mathop{\rm log} \ 
K _  \epsilon  ( X, \rho _ {n} ),
$$

where  $  \rho _ {n} ( x, y) = \max _ {0 \leq  i \leq  n }   \rho ( S  ^ {i} x, S  ^ {i} y) $(
cf. [[#References|[2]]]–[[#References|[4]]]).

It turns out that

$$ 
h _ { \mathop{\rm top}  } ( S  ^ {n} )  = \ 
n h _ { \mathop{\rm top}  } ( S),
$$

and if  $  S $
is a homeomorphism, then  $  h _ { \mathop{\rm top}  } ( S  ^ {-} 1 ) = h _ { \mathop{\rm top}  } ( S) $. 
Hence it is natural to take the topological entropy of a [[Cascade|cascade]]  $  \{ S  ^ {n} \} $
to be  $  h _ { \mathop{\rm top}  } ( S) $. 
For a topological [[Flow (continuous-time dynamical system)|flow (continuous-time dynamical system)]]  $  \{ S _ {t} \} $
it turns out that

$$ 
h _ { \mathop{\rm top}  } ( S _ {t} )  = \ 
| t |  h _ { \mathop{\rm top}  } ( S _ {1} ),
$$

so it is natural to take the topological entropy of the flow to be  $  h _ { \mathop{\rm top}  } ( S _ {1} ) $. 
In a somewhat different way one can define the topological entropy for other transformation groups (it no longer reduces to the topological entropy of one of the elements of the group; cf. [[#References|[7]]]).

The topological entropy  $  h _ { \mathop{\rm top}  } ( S) $
coincides with the supremum of the metric entropy  $  h _  \mu  ( S) $
over all possible normalized invariant Borel measures  $  \mu $(
cf. [[#References|[2]]], [[#References|[5]]]–[[#References|[7]]]; [[#References|[18]]])

for the existence of  $  \max  h _  \mu  $
and the dependence of  $  h _ { \mathop{\rm top}  } ( S) $
on  $  S $). 
This is a special case of the variational principle, which establishes a topological interpretation of the value

$$ 
\sup _  \mu   \left [ h _  \mu  ( S) + \int\limits f  d \mu \right ]
$$

for a fixed continuous function  $  f $(
cf. [[#References|[4]]], [[#References|[8]]], [[#References|[9]]]). The topological entropy gives a characteristic of the  "complexity"  or  "diversity"  of motions in a dynamical system (cf. [[#References|[10]]], [[#References|[3]]], [[#References|[4]]]). It is also connected in certain cases with the asymptotics (as  $  T \rightarrow \infty $) 
of the number of periodic trajectories (of period  $  \leq  T $; 
cf. [[Periodic trajectory|Periodic trajectory]] and [[#References|[3]]], [[#References|[4]]], [[#References|[11]]]–[[#References|[13]]]). The  "entropy conjecture"  asserts that the topological entropy of a diffeomorphism  $  S $
of a closed manifold  $  W $
is not less than the logarithm of the [[Spectral radius|spectral radius]] of the linear transformation induced by  $  S $
on the homology spaces  $  H _ {*} ( W;  \mathbf R ) $(
cf. [[#References|[14]]],[[#References|[15]]]). It has been proved in the  $  C  ^  \infty  $-
case, .

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  R.L. Adler,   A.G. Konheim,   M.H. McAndrew,   "Topological entropy"  ''Trans. Amer. Math. Soc.'' , '''114'''  (1965)  pp. 309–319</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  E.I. Dinaburg,   "On the relation between various entropy characteristics of dynamical systems"  ''Math. USSR Izv.'' , '''5''' :  2  (1971)  pp. 337–378  ''Izv. Akad. Nauk SSSR, Ser. Mat.'' , '''35''' :  2  (1971)  pp. 324–366</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  V.M. Alekseev,   "Symbolic dynamics" , ''Eleventh Mathematical Summer School'' , Kiev  (1976)  pp. 5–210  (In Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  R. Boywen,   "Methods of symbolic dynamics" , Moscow  (1979)  (In Russian; translated from English)  (Collection of papers)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  T.N.T. Goodman,   "Relating topological entropy and measure entropy"  ''Bull. London Math. Soc.'' , '''3'''  (1971)  pp. 176–180</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  L.W. Goodwyn,   "Comparing topological entropy with measure-theoretic entropy"  ''Amer. J. Math.'' , '''94'''  (1972)  pp. 366–368</TD></TR><TR><TD valign="top">[7a]</TD> <TD valign="top">  A.T. Tagi-zade,   "The entropy of motions of amenable groups"  ''Dokl. Akad. Nauk AzerbSSR'' , '''34''' :  6  (1978)  pp. 18–22  (In Russian)</TD></TR><TR><TD valign="top">[7b]</TD> <TD valign="top">  A.T. Tagi-zade,   "Entropy characteristics of amenable groups"  ''Dokl. Akad. Nauk AzerbSSR'' , '''34''' :  8  (1978)  pp. 11–14  (In Russian)</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top">  A.M. Stepin,   A.T. Tagi-zade,   "Variational characterization of the topological pressure of amenable transformation groups"  ''Soviet Math. Dokl.'' , '''22''' :  2  (1980)  pp. 405–409  ''Dokl. Akad. Nauk SSSR'' , '''254''' :  3  (1980)  pp. 545–548</TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top">  J. Moulin Ollagnier,   D. Pinchon,   "The variational principle"  ''Studia Math.'' , '''72'''  (1982)  pp. 151–159</TD></TR><TR><TD valign="top">[10]</TD> <TD valign="top">  A.A. Brudno,   "Entropy and the complexity of the trajectories of a dynamical system"  ''Trans. Moscow Math. Soc.'' , '''44'''  (1982)  pp. 127–152  ''Trudy Moskov. Mat. Obshch.'' , '''44'''  (1982)  pp. 124–149</TD></TR><TR><TD valign="top">[11a]</TD> <TD valign="top">  A.G. Kushnirenko,   "Problems in the general theory of dynamical systems on a manifold"  ''Transl. Amer. Math. Soc.'' , '''116'''  (1981)  pp. 1–42  ''Ninth Math. Summer School''  (1976)  pp. 52–124</TD></TR><TR><TD valign="top">[11b]</TD> <TD valign="top">  A.B. Katok,   "Dynamical systems with hyperbolic structure"  ''Transl. Amer. Math. Soc.'' , '''116'''  (1981)  pp. 43–96  ''Ninth Math. Summer School''  (1976)  pp. 125–211</TD></TR><TR><TD valign="top">[11c]</TD> <TD valign="top">  V.M. Alekseev,   "Quasirandom oscillations and qualitative questions in celestial mechanics"  ''Transl. Amer. Math. Soc.'' , '''116'''  (1981)  pp. 97–169  ''Ninth Math. Summer School''  (1976)  pp. 212–341</TD></TR><TR><TD valign="top">[12]</TD> <TD valign="top">  A.B. Katok,   Ya.G. Sinai,   A.M. Stepin,   "The theory of dynamical systems and general transformation groups with invariant measure"  ''J. Soviet Math.'' , '''7''' :  6  (1977)  pp. 974–1065  ''Itogi Nauk. i Tekhn. Mat. Anal.'' , '''13'''  (1975)  pp. 129–262</TD></TR><TR><TD valign="top">[13]</TD> <TD valign="top">  A.B. Katok,   "Lyapunov exponents, entropy and periodic orbits for diffeomorphisms"  ''Publ. Math. IHES'' , '''51'''  (1980)  pp. 137–173</TD></TR><TR><TD valign="top">[14a]</TD> <TD valign="top">  A.B. Katok,   "The entropy conjecture"  D.V. Anosov (ed.) , ''Smooth dynamical systems'' , Moscow  (1977)  pp. 181–203  (In Russian)</TD></TR><TR><TD valign="top">[14b]</TD> <TD valign="top">  M. Shub,   "Dynamical systems, filtrations and entropy"  ''Bull. Amer. Math. Soc.'' , '''80'''  (1974)  pp. 27–41</TD></TR><TR><TD valign="top">[15]</TD> <TD valign="top">  D. Fried,   M. Shub,   "Entropy, linearity and chain-recurrence"  ''Publ. Math. IHES'' , '''50'''  (1979)  pp. 203–214</TD></TR><TR><TD valign="top">[16]</TD> <TD valign="top">  C. Grilleneberger,   "Ergodic theory on compact spaces" , Springer  (1976)</TD></TR></table>

====Comments====
Instead of the collection of translations [[#References|[4]]] one may consult [[#References|[a2]]].

In the above,  $  h _  \mu  ( S) $
denotes the entropy of  $  S $
with respect to the normalized invariant Borel measure  $  \mu $(
cf. [[Entropy theory of a dynamical system|Entropy theory of a dynamical system]]). The value  $  P _ {S} ( f  ) = \sup _  \mu  [ h _  \mu  ( S) + \int f  d \mu ] $, 
where  $  \mu $
runs over the set of all normalized invariant Borel measures, is called the pressure of  $  f $(
with respect to  $  S $). 
If  $  \mu $
satisfies  $  h _  \mu  ( S) + \int f  d \mu = P _ {S} ( f  ) $(
i.e., the sup is a max), then  $  \mu $
is called an equilibrium state or Gibbs measure for  $  f $(
with respect to  $  S $). 
See [[#References|[a2]]], also for existence and uniqueness results.

For recent results about the estimation of topological entropy, see [[#References|[a1]]] and the references given there.

====References====
<table>
<TR><TD valign="top">[a1]</TD> <TD valign="top">  S.E. Newhouse,   "Entropy and volume"  ''Ergod. Th. &amp; Dynam. Syst.'' , '''8'''  (1988)  pp. 283–299  {{ZBL|0638.58016}}</TD></TR>
<TR><TD valign="top">[a2]</TD> <TD valign="top">  R. Bowen,   "Equilibrium states and the ergodic theory of Anosov diffeomorphisms" , ''Lect. notes in math.'' , '''470''' , Springer  (1975)  {{ZBL|0308.28010}}; 2nd ed.
 (2008) ISBN 978-3-540-77605-5  {{ZBL|1172.37001}}</TD></TR>
</table>