|
|
| Line 1: |
Line 1: |
| [[Image:Interval exchange.svg|thumb|right|Graph of interval exchange transformation (in black) with <math>\lambda = (1/15,2/15,3/15,4/15,5/15)</math> and <math>\pi=(3,5,2,4,1)</math>. In blue, the orbit generated starting from <math>1/2</math>.]]
| | Marvella is what you can contact her but it's not the most feminine title out there. Minnesota is exactly where he's been residing for years. Hiring is my profession. What I adore doing is doing ceramics but I haven't produced a dime with it.<br><br>Look at my blog post over the counter std test - [http://rivoli.enaiponline.com/user/view.php?id=438251&course=1 linked internet site], |
| In [[mathematics]], an '''interval exchange transformation'''<ref>Michael Keane, ''Interval exchange transformations'', Mathematische Zeitschrift 141, 25 (1975), ''http://www.springerlink.com/content/q10w48161l15gg18/''</ref> 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 <math>n > 0</math> and let <math>\pi</math> be a [[permutation]] on <math>1, \dots, n</math>. Consider a [[Vector (geometric)|vector]] <math>\lambda = (\lambda_1, \dots, \lambda_n)</math> of positive real numbers (the widths of the subintervals), satisfying
| |
| | |
| :<math>\sum_{i=1}^n \lambda_i = 1.</math>
| |
| | |
| Define a map <math>T_{\pi,\lambda}:[0,1]\rightarrow [0,1],</math> called the '''interval exchange transformation associated to the pair <math>(\pi,\lambda)</math>''' as follows. For <math>1 \leq i \leq n</math> let
| |
| | |
| :<math>a_i = \sum_{1 \leq j < i} \lambda_j \quad \text{and} \quad a'_i = \sum_{1 \leq j < \pi(i)} \lambda_{\pi^{-1}(j)}.</math>
| |
| | |
| Then for <math>x \in [0,1]</math>, define
| |
| | |
| :<math>
| |
| T_{\pi,\lambda}(x) = x - a_i + a'_i
| |
| </math>
| |
| | |
| if <math>x</math> lies in the subinterval <math>[a_i,a_i+\lambda_i)</math>. Thus <math>T_{\pi,\lambda}</math> acts on each subinterval of the form <math>[a_i,a_i+\lambda_i)</math> by a [[translation (geometry)|translation]], and it rearranges these subintervals so that the subinterval at position <math>i</math> is moved to position <math>\pi(i)</math>.
| |
| | |
| ==Properties==
| |
| Any interval exchange transformation <math>T_{\pi,\lambda}</math> is a [[bijection]] of <math>[0,1]</math> to itself preserves the [[Lebesgue measure]]. It is continuous except at a finite number of points.
| |
| | |
| The [[Inverse function|inverse]] of the interval exchange transformation <math>T_{\pi,\lambda}</math> is again an interval exchange transformation. In fact, it is the transformation <math>T_{\pi^{-1}, \lambda'}</math> where <math>\lambda'_i = \lambda_{\pi^{-1}(i)}</math> for all <math>1 \leq i \leq n</math>.
| |
| | |
| If <math>n=2</math> and <math>\pi = (12)</math> (in [[cycle notation]]), and if we join up the ends of the interval to make a circle, then <math>T_{\pi,\lambda}</math> is just a circle rotation. The [[Weyl equidistribution theorem]] then asserts that if the length <math>\lambda_1</math> is [[irrational]], then <math>T_{\pi,\lambda}</math> is [[uniquely ergodic]]. Roughly speaking, this means that the orbits of points of <math>[0,1]</math> are uniformly evenly distributed. On the other hand, if <math>\lambda_1</math> is rational then each point of the interval is [[Frequency|periodic]], and the period is the denominator of <math>\lambda_1</math> (written in lowest terms).
| |
| | |
| If <math>n>2</math>, and provided <math>\pi</math> satisfies certain non-degeneracy conditions (namely there is no integer <math>0 < k < n</math> such that <math>\pi(\{1,\dots,k\}) = \{1,\dots,k\}</math>), a deep theorem which was a conjecture of M.Keane and due independently to W.Veech <ref>William A. Veech, ''Gauss measures for transformations on the space of interval exchange maps'', Annals of Mathematics 115 (1982), ''http://www.jstor.org/stable/1971391''</ref> and to H.Masur <ref>Howard Masur, ''Interval Exchange Transformations and Measured Foliations'', Annals of Mathematics 115 (1982) ''http://www.jstor.org/stable/1971341''</ref> asserts that for [[almost all]] choices of <math>\lambda</math> in the unit simplex <math>\{(t_1, \dots, t_n) : \sum t_i = 1\}</math> the interval exchange transformation <math>T_{\pi,\lambda}</math> is again [[uniquely ergodic]]. However, for <math>n \geq 4</math> there also exist choices of <math>(\pi,\lambda)</math> so that <math>T_{\pi,\lambda}</math> is [[ergodic]] but not [[uniquely ergodic]]. Even in these cases, the number of ergodic [[Invariant (mathematics)|invariant]] [[measures]] of <math>T_{\pi,\lambda}</math> is finite, and is at most <math>n</math>.
| |
| | |
| ==Generalizations==
| |
| Two and higher-dimensional generalizations include polygon exchanges, polyhedral exchanges and [[Piecewise isometry|Piecewise isometries]]<ref>[http://math.sfsu.edu/goetz/Research/graz/graz.pdf Piecewise isometries - an emerging area of dynamical systems], Arek Goetz</ref>
| |
| | |
| ==Notes== | |
| {{reflist}}
| |
| | |
| == References ==
| |
| * Artur Avila and Giovanni Forni, ''Weak mixing for interval exchange transformations and translation flows'', arXiv:math/0406326v1, ''http://arxiv.org/abs/math.DS/0406326''
| |
| | |
| {{Chaos theory}}
| |
| | |
| [[Category:Chaotic maps]]
| |
Marvella is what you can contact her but it's not the most feminine title out there. Minnesota is exactly where he's been residing for years. Hiring is my profession. What I adore doing is doing ceramics but I haven't produced a dime with it.
Look at my blog post over the counter std test - linked internet site,