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{{MSC|37A05}}
 
{{MSC|37A05}}
  
 
[[Category:Measure-preserving transformations]]
 
[[Category:Measure-preserving transformations]]
  
One of the methods of [[Ergodic theory|ergodic theory]]. Any automorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012900/a0129001.png" /> of a Lebesgue space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012900/a0129002.png" /> with measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012900/a0129003.png" /> can be obtained as the limit of periodic automorphisms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012900/a0129004.png" /> in the natural weak or uniform topology of the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012900/a0129005.png" /> of all automorphisms {{Cite|H}}. To characterize the rate of approximation quantitatively one considers not only the automorphisms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012900/a0129006.png" />, but also finite measurable decompositions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012900/a0129007.png" /> which are invariant under <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012900/a0129008.png" />, i.e. decompositions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012900/a0129009.png" /> into a finite number of non-intersecting measurable sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012900/a01290010.png" />, which are mapped into each other by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012900/a01290011.png" />. The number
+
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
  
<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/a/a012/a012900/a01290012.png" /></td> </tr></table>
+
$$
 +
d ( T , T _ {n} ; \xi _ {n} )  = \
 +
\sum _ {i = 1 } ^ { {q } _ {n} }
 +
\mu ( T C _ {n,i}  \Delta  T _ {n} C _ {n,i} )
 +
$$
  
is an estimate of the proximity of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012900/a01290013.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012900/a01290014.png" /> with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012900/a01290015.png" />; here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012900/a01290016.png" /> is the symmetric difference
+
is an estimate of the proximity of $  T _ {n} $
 +
to $  T $
 +
with respect to $  \xi _ {n} $;  
 +
here $  \Delta $
 +
is the symmetric difference
  
<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/a/a012/a012900/a01290017.png" /></td> </tr></table>
+
$$
 +
A \Delta B  = ( A \setminus B ) \cup ( B \setminus  A ).
 +
$$
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012900/a01290018.png" /> is given, it is possible to choose <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012900/a01290019.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012900/a01290020.png" /> (with the above properties) such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012900/a01290021.png" /> is arbitrarily small {{Cite|H}}. The metric invariants of the automorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012900/a01290022.png" /> become apparent on considering infinite sequences <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012900/a01290023.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012900/a01290024.png" /> such that for any measurable set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012900/a01290025.png" /> there exists a sequence of sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012900/a01290026.png" />, each being the union of some of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012900/a01290027.png" />, which approximates <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012900/a01290028.png" /> in the sense that
+
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
  
<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/a/a012/a012900/a01290029.png" /></td> </tr></table>
+
$$
 +
\lim\limits _ {n \rightarrow \infty }  \mu ( A \Delta A _ {n} )  = 0
 +
$$
  
( "the decompositions xn converge to a decomposition into points" ). If, in addition, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012900/a01290030.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012900/a01290031.png" /> is a given monotone sequence tending to zero, then one says that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012900/a01290032.png" /> admits an approximation of the first type by periodic transformations with rate <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012900/a01290033.png" />; if, in addition, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012900/a01290034.png" /> permutes the sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012900/a01290035.png" /> cyclically, then one speaks of cyclic approximation by periodic transformations. For other variants see {{Cite|KS}}, {{Cite|ACS}}, {{Cite|S}}.
+
( "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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012900/a01290036.png" /> affect the properties of the limit automorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012900/a01290037.png" />. Thus, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012900/a01290038.png" /> has a cyclic approximation by periodic transformations with a rate <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012900/a01290039.png" />, then, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012900/a01290040.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012900/a01290041.png" /> will be ergodic; if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012900/a01290042.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012900/a01290043.png" /> will not be mixing; and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012900/a01290044.png" />, the spectrum of the corresponding unitary shift operator is simple. Certain properties of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012900/a01290045.png" /> may be described in terms of the rate of approximation. Thus, its [[Entropy|entropy]] is equal to the lower bound of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012900/a01290046.png" />'s for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012900/a01290047.png" /> admits an approximation by periodic transformations of the first kind with a rate of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012900/a01290048.png" /> {{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}}.
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012900/a01290049.png" />, provided with the weak topology, may be considerably strengthened: For any monotone sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012900/a01290050.png" />, the automorphisms which allow cyclic approximations at the rate <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012900/a01290051.png" /> form a set of the second category in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012900/a01290052.png" /> {{Cite|KS}}. Accordingly, approximations by periodic transformations yield so-called category theorems, which state that in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012900/a01290053.png" /> (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}}).
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012900/a01290054.png" /> be a topological or smooth manifold, and let the measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012900/a01290055.png" /> be compatible with the topology or with the differential structure. In the class of homeomorphisms or diffeomorphisms preserving <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012900/a01290056.png" /> it is not the weak topology, but other topologies that are natural. Category theorems analogous to those for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012900/a01290057.png" /> are valid for homeomorphisms; for the history of the problem and its present "state-of-the-art" see {{Cite|KS2}}.
+
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====
 
====References====
Line 45: Line 119:
 
Contributions to the foundation of the theory of approximations were also made by V.A. Rokhlin (cf. {{Cite|R}}).
 
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012900/a01290058.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012900/a01290059.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012900/a01290060.png" /> is periodic of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012900/a01290061.png" />,
+
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} $,
  
<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/a/a012/a012900/a01290062.png" /></td> </tr></table>
+
$$
 +
\sum _ { i=1 } ^ { {q } _ {n} }
 +
\mu ( T C _ {n,i} \
 +
\Delta  T _ {n} C _ {n,i} )
 +
< f ( q _ {n} )
 +
$$
  
and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012900/a01290063.png" /> in the strong topology for operators on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012900/a01290064.png" />, then one says that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012900/a01290065.png" /> admits an approximation of the second type by periodic transformations with speed <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012900/a01290066.png" />. Reference {{Cite|CFS}} is a basic and well-known one.
+
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.
 
In this context one also speaks of partitions instead of decompositions.

Revision as of 18:47, 5 April 2020


2020 Mathematics Subject Classification: Primary: 37A05 [MSN][ZBL]

One of the methods of 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 [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 [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 [KS], [ACS], [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 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 $[KS], [S]. Approximations by periodic transformations were used in the study of a number of simple examples [KS], including smooth flows on two-dimensional surfaces [Ko]. They served in the construction of a number of dynamical systems with unexpected metric properties [KS], [ACS], [S], or with an unexpected combination of metric and differential properties [AK], [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 $[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 [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 [KS2].

References

[H] P.R. Halmos, "Lectures on ergodic theory" , Math. Soc. Japan (1956) MR0097489 Zbl 0073.09302
[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 MR0219697 Zbl 0172.07202
[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 MR0370662 Zbl 0255.58007
[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 MR0331425 Zbl 0316.58010
[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 Zbl 0209.27803
[ACS] M.A. Akcoglu, R.V. Chacon, T. Schwartzbauer, "Commuting transformations and mixing" Proc. Amer. Math. Soc. , 24 pp. 637–642 MR0254212 Zbl 0197.04001
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Comments

Contributions to the foundation of the theory of approximations were also made by V.A. Rokhlin (cf. [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 [CFS] is a basic and well-known one.

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

References

[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) MR832433
[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 Zbl 0185.21802
How to Cite This Entry:
Approximation by periodic transformations. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Approximation_by_periodic_transformations&oldid=45198
This article was adapted from an original article by D.V. Anosov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article