Approximation by periodic transformations

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
[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

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