|
Interval Exchange Transformation
In mathematics, an interval exchange transformation 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. They arise naturally in the study of polygonal billiards and in area-preserving flows. Formal definition Let n > 0 and let \pi be a permutation on 1, \dots, n. Consider a vector \lambda = (\lambda_1, \dots, \lambda_n) of positive real numbers (the widths of the subintervals), satisfying :\sum_^n \lambda_i = 1. Define a map T_: ,1rightarrow ,1 called the interval exchange transformation associated with the pair (\pi,\lambda) as follows. For 1 \leq i \leq n let :a_i = \sum_ \lambda_j \quad \text \quad a'_i = \sum_ \lambda_. Then for x \in ,1/math>, define : T_(x) = x - a_i + a'_i if x lies in the subinterval translation, and it rearranges these subintervals so that the subinterval at position i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Interval Exchange
In mathematics, an interval exchange transformation is a kind of dynamical system that generalises Irrational rotation, 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. They arise naturally in the study of Dynamical billiards, polygonal billiards and in area-preserving flows. Formal definition Let n > 0 and let \pi be a permutation on 1, \dots, n. Consider a Vector (geometric), vector \lambda = (\lambda_1, \dots, \lambda_n) of positive real numbers (the widths of the subintervals), satisfying :\sum_^n \lambda_i = 1. Define a map T_:[0,1]\rightarrow [0,1], called the interval exchange transformation associated with the pair (\pi,\lambda) as follows. For 1 \leq i \leq n let :a_i = \sum_ \lambda_j \quad \text \quad a'_i = \sum_ \lambda_. Then for x \in [0,1], define : T_(x) = x - a_i + a'_i if x lies in the subinterval [a_i,a_i+\lambda_i). Thus ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
