# 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].

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