“Interval exchange transformation”的意思、由来-开放百科全书

In mathematics, an interval exchange transformation^[1] is a kind of dynamical system that generalises circle rotation. The phase space consists of the unit interval, and the transformation acts by cutting the interval into several subintervals, and then permuting these subintervals.

Formal definition

Let

and let

be a permutation on

. Consider a vector

of positive real numbers (the widths of the subintervals), satisfying

Define a map

called the interval exchange transformation associated with the pair as follows. For

let

lies in the subinterval

. Thus

acts on each subinterval of the form

by a translation, and it rearranges these subintervals so that the subinterval at position

is moved to position

Properties

Any interval exchange transformation

is a bijection of

to itself preserves the Lebesgue measure. It is continuous except at a finite number of points.

The inverse of the interval exchange transformation

is again an interval exchange transformation. In fact, it is the transformation

where

for all

and

(in cycle notation), and if we join up the ends of the interval to make a circle, then

is just a circle rotation. The Weyl equidistribution theorem then asserts that if the length

is irrational, then

is uniquely ergodic. Roughly speaking, this means that the orbits of points of

are uniformly evenly distributed. On the other hand, if

is rational then each point of the interval is periodic, and the period is the denominator of

(written in lowest terms).

, and provided

satisfies certain non-degeneracy conditions (namely there is no integer

such that

), a deep theorem which was a conjecture of M.Keane and due independently to William A. Veech^[2] and to Howard Masur ^[3] asserts that for almost all choices of

in the unit simplex

the interval exchange transformation

is again uniquely ergodic. However, for

there also exist choices of

so that

is ergodic but not uniquely ergodic. Even in these cases, the number of ergodic invariant measures of

is finite, and is at most

Generalizations

Two and higher-dimensional generalizations include polygon exchanges, polyhedral exchanges and piecewise isometries.^[4]

Notes

1. ^{{citation | last = Keane | first = Michael | doi = 10.1007/BF01236981 | journal = Mathematische Zeitschrift | mr = 0357739 | pages = 25–31 | title = Interval exchange transformations | volume = 141 | year = 1975}}.
2. ^{{citation | last = Veech | first = William A. | authorlink = William A. Veech | doi = 10.2307/1971391 | issue = 1 | journal = Annals of Mathematics | mr = 644019 | pages = 201–242 | series = Second Series | title = Gauss measures for transformations on the space of interval exchange maps | volume = 115 | year = 1982}}.
3. ^{{citation | last = Masur | first = Howard | doi = 10.2307/1971341 | issue = 1 | journal = Annals of Mathematics | mr = 644018 | pages = 169–200 | series = Second Series | title = Interval exchange transformations and measured foliations | volume = 115 | year = 1982}}.
4. ^Piecewise isometries – an emerging area of dynamical systems, Arek Goetz

Formal definition

Properties

Generalizations

Notes

References