# Approximation by periodic transformations

2010 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 [S] T. Schwartzbauer, "Entropy and approximation of measure preserving transformations" Pacific J. Math. , 43 (1972) pp. 753–764 MR0316683 Zbl 0259.28012 [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 MR0516507

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=52269
This article was adapted from an original article by D.V. Anosov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article