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{{MSC|37A05}}

[[Category:Measure-preserving transformations]]

One of the methods of [[Ergodic theory|ergodic theory]]. Any automorphism  $  T $
of a Lebesgue space  $  X $
with measure  $  \mu $
can be obtained as the limit of periodic automorphisms  $  T _ {n} $
in the natural weak or uniform topology of the space  $  \mathfrak A $
of all automorphisms {{Cite|H}}. To characterize the rate of approximation quantitatively one considers not only the automorphisms  $  T _ {n} $, 
but also finite measurable decompositions of  $  X $
which are invariant under  $  T _ {n} $, 
i.e. decompositions of  $  X $
into a finite number of non-intersecting measurable sets  $  C _ {n,1 }  \dots C _ {n, q _ {n}  } $, 
which are mapped into each other by  $  T _ {n} $. 
The number

$$ 
d ( T , T _ {n} ;  \xi _ {n} )  = \ 
\sum _ {i = 1 } ^ { {q } _ {n} }
\mu ( T C _ {n,i}  \Delta  T _ {n} C _ {n,i} )
$$

is an estimate of the proximity of  $  T _ {n} $
to  $  T $
with respect to  $  \xi _ {n} $; 
here  $  \Delta $
is the symmetric difference

$$ 
A \Delta B  =  ( A \setminus B ) \cup ( B \setminus  A ).
$$

If  $  q _ {n} $
is given, it is possible to choose  $  \xi _ {n} $
and  $  T _ {n} $ (with the above properties) such that  $  d (T, T _ {n} ;  \xi _ {n} ) $
is arbitrarily small {{Cite|H}}. The metric invariants of the automorphism  $  T $
become apparent on considering infinite sequences  $  T _ {n} $
and  $  \xi _ {n} $
such that for any measurable set  $  A $
there exists a sequence of sets  $  A _ {n} $, 
each being the union of some of the  $  C _ {n,i }  $, 
which approximates  $  A $
in the sense that

$$ 
\lim\limits _ {n \rightarrow \infty }  \mu ( A \Delta A _ {n} )  =  0
$$

( "the decompositions xn converge to a decomposition into points" ). If, in addition,  $  d(T, T _ {n} ;  \xi _ {n} ) < f (q _ {n} ) $, 
where  $  f(n) $
is a given monotone sequence tending to zero, then one says that  $  T $
admits an approximation of the first type by periodic transformations with rate  $  f(n) $; 
if, in addition,  $  T _ {n} $
permutes the sets  $  C _ {n,i }  $
cyclically, then one speaks of cyclic approximation by periodic transformations. For other variants see {{Cite|KS}}, {{Cite|ACS}}, {{Cite|S}}.

At a certain rate of approximation certain properties of the periodic automorphisms  $  T _ {n} $
affect the properties of the limit automorphism  $  T $. 
Thus, if  $  T $
has a cyclic approximation by periodic transformations with a rate  $  c/n $, 
then, if  $  c < 4 $, 
$  T $
will be ergodic; if  $  c < 2 $, 
$  T $
will not be mixing; and if  $  c < 1 $, 
the spectrum of the corresponding unitary shift operator is simple. Certain properties of  $  T $
may be described in terms of the rate of approximation. Thus, its [[Entropy|entropy]] is equal to the lower bound of the  $  c $'
s for which  $  T $
admits an approximation by periodic transformations of the first kind with a rate of  $  2c /  \mathop{\rm log} _ {2}  n ${{
Cite|KS}}, {{Cite|S}}. Approximations by periodic transformations were used in the study of a number of simple examples {{Cite|KS}}, including smooth flows on two-dimensional surfaces {{Cite|Ko}}. They served in the construction of a number of dynamical systems with unexpected metric properties {{Cite|KS}}, {{Cite|ACS}}, {{Cite|S}}, or with an unexpected combination of metric and differential properties {{Cite|AK}}, {{Cite|Ka}}.

The statement on the density of periodic automorphisms in  $  \mathfrak A $, 
provided with the weak topology, may be considerably strengthened: For any monotone sequence  $  f(n) > 0 $, 
the automorphisms which allow cyclic approximations at the rate  $  f(n) $
form a set of the second category in  $  \mathfrak A ${{
Cite|KS}}. Accordingly, approximations by periodic transformations yield so-called category theorems, which state that in  $  \mathfrak A $ (with the weak topology) the automorphisms with a given property form a set of the first or second category (e.g. ergodic sets are of the second category, while mixing sets are of the first category {{Cite|H}}).

Let  $  X $
be a topological or smooth manifold, and let the measure  $  \mu $
be compatible with the topology or with the differential structure. In the class of homeomorphisms or diffeomorphisms preserving  $  \mu $
it is not the weak topology, but other topologies that are natural. Category theorems analogous to those for  $  \mathfrak A $
are valid for homeomorphisms; for the history of the problem and its present "state-of-the-art" see {{Cite|KS2}}.

====References====
{|
|valign="top"|{{Ref|H}}|| P.R. Halmos, "Lectures on ergodic theory" , Math. Soc. Japan (1956) {{MR|0097489}} {{ZBL|0073.09302}}
|-
|valign="top"|{{Ref|KS}}|| A.B. Katok, A.M. Stepin, "Approximations in ergodic theory" ''Russian Math. Surveys'' , '''22''' : 5 (1967) pp. 77–102 ''Uspekhi Mat. Nauk'' , '''22''' : 5 (1967) pp. 81–106 {{MR|0219697}} {{ZBL|0172.07202}}
|-
|valign="top"|{{Ref|AK}}|| D.V. Anosov, A.B. Katok, "New examples in smooth ergodic theory. Ergodic diffeomorphisms" ''Trans. Moscow Math. Soc.'' , '''23''' (1970) pp. 3–36 ''Trudy Moskov. Mat. Obshch.'' , '''23''' (1970) pp. 1–35 {{MR|0370662}} {{ZBL|0255.58007}}
|-
|valign="top"|{{Ref|Ka}}|| A.B. Katok, "Ergodic perturbations of degenerate integrable Hamiltonian systems" ''Math. USSR-Izv.'' , '''7''' : 3 (1973) pp. 535–571 ''Izv. Akad. Nauk SSSR Ser. Mat.'' , '''37''' (1973) pp. 539–576 {{MR|0331425}} {{ZBL|0316.58010}}
|-
|valign="top"|{{Ref|KS2}}|| A.B. Katok, A.M. Stepin, "Metric properties of measure preserving homeomorphisms" ''Russian Math. Surveys'' , '''25''' : 2 (1970) pp. 191–220 ''Uspekhi Mat. Nauk'' , '''25''' : 2 (1970) pp. 193–220 {{MR|}} {{ZBL|0209.27803}}
|-
|valign="top"|{{Ref|ACS}}|| M.A. Akcoglu, R.V. Chacon, T. Schwartzbauer, "Commuting transformations and mixing" ''Proc. Amer. Math. Soc.'' , '''24''' pp. 637–642 {{MR|0254212}} {{ZBL|0197.04001}}
|-
|valign="top"|{{Ref|S}}|| T. Schwartzbauer, "Entropy and approximation of measure preserving transformations" ''Pacific J. Math.'' , '''43''' (1972) pp. 753–764 {{MR|0316683}} {{ZBL|0259.28012}}
|-
|valign="top"|{{Ref|Ko}}|| A.V. Kochergin, "On mixing in special flows over a shifting of segments and in smooth flows on surfaces" ''Math. USSR-Sb.'' , '''25''' : 3 (1975) pp. 441–469 ''Mat. Sb.'' , '''96''' : 3 (1975) pp. 472–502 {{MR|0516507}}
|}

====Comments====
Contributions to the foundation of the theory of approximations were also made by V.A. Rokhlin (cf. {{Cite|R}}).

If in an approximation by periodic transformations one has the following inequality for the sequences  $  \{ \xi _ {n} \} $, 
$  \{ T _ {n} \} $, 
where  $  T _ {n} $
is periodic of order  $  q _ {n} $,

$$ 
\sum _ { i=1 } ^ { {q } _ {n} }
\mu ( T C _ {n,i} \ 
\Delta  T _ {n} C _ {n,i} )
 <  f ( q _ {n} )
$$

and  $  U _ {T _ {n}  } \rightarrow U _ {T} $
in the strong topology for operators on  $  L _ {2} ( X , \mu ) $, 
then one says that  $  T $
admits an approximation of the second type by periodic transformations with speed  $  f (n) $. 
Reference {{Cite|CFS}} is a basic and well-known one.

In this context one also speaks of partitions instead of decompositions.

====References====
{|
|valign="top"|{{Ref|CFS}}|| I.P. [I.P. Kornfel'd] Cornfel'd, S.V. Fomin, Ya.G. Sinai, "Ergodic theory" , Springer (1982) pp. Chapt. 15;16 (Translated from Russian) {{MR|832433}} {{ZBL|}}
|-
|valign="top"|{{Ref|R}}|| V.A. Rokhlin, "Selected topics from the metric theory of dynamical systems" ''Amer. Math. Soc. Transl. Series 2'' , '''49''' pp. 171–240 ''Uspekhi Mat. Nauk'' , '''4''' : 2 (30) (1949) pp. 57–128 {{MR|}} {{ZBL|0185.21802}}
|}