View source for Approximation by periodic transformations

{{MSC|37A05}}

[[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 [[#References|[1]]]. 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

<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>

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

<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>

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 [[#References|[1]]]. 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

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( "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 [[#References|[2]]], [[#References|[6]]], [[#References|[7]]].

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" /> [[#References|[2]]], [[#References|[7]]]. Approximations by periodic transformations were used in the study of a number of simple examples [[#References|[2]]], including smooth flows on two-dimensional surfaces [[#References|[8]]]. They served in the construction of a number of dynamical systems with unexpected metric properties [[#References|[2]]], [[#References|[6]]], [[#References|[7]]], or with an unexpected combination of metric and differential properties [[#References|[3]]], [[#References|[4]]].

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" /> [[#References|[2]]]. 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 [[#References|[1]]]).

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 [[#References|[5]]].

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> P.R. Halmos, "Lectures on ergodic theory" , Math. Soc. Japan (1956) {{MR|0097489}} {{ZBL|0073.09302}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> 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}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> 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}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> 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}} </TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> 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}} </TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top"> 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}} </TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top"> T. Schwartzbauer, "Entropy and approximation of measure preserving transformations" ''Pacific J. Math.'' , '''43''' (1972) pp. 753–764 {{MR|0316683}} {{ZBL|0259.28012}} </TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top"> 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}}</TD></TR></table>

====Comments====
Contributions to the foundation of the theory of approximations were also made by V.A. Rokhlin (cf. [[#References|[a2]]]).

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" />,

<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>

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 [[#References|[a1]]] is a basic and well-known one.

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

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> 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|}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> 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}} </TD></TR></table>