Invariant distributions for interval exchange transformations
type: Item
instance of: scholarly article
title: Invariant distributions for interval exchange transformations
zbMATH Open document ID: 1264.32008
author: William A. Veech
published in: Dynamical Systems and Group Actions
publication date: 2012-09-05T00:00:00Z
Mathematics Subject Classification ID: 32G15
Mathematics Subject Classification ID: 30F60
Mathematics Subject Classification ID: 37D40
Mathematics Subject Classification ID: 37F30
zbMATH DE Number: 6077504
zbMATH Keywords: invariant distributions for interval exchange maps
zbMATH Keywords: Forni cocycle
zbMATH Keywords: Bufetov stratum
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1090/conm/567/11254
OpenAlex ID: W4211259895
DOI: 10.1090/CONM/567/11254
review text: The purpose of this paper is to point out that the coboundary equation from [\textit{W. A. Veech}, Am. J. Math. 106, 1389--1422 (1984; Zbl 0631.28008); ibid. 106, 1361--1387 (1984; Zbl 0631.28007); ibid. 106, 1331--1359 (1984; Zbl 0631.28006)] leads naturally to spaces of invariant distributions for interval exchange maps. The author is motivated by A. Bufetov's insightful characterization of fibers relative to Forni's Lyapunov splitting of the Forni cocycle. For a generic point in a stratum, Bufetov's characterization is in terms of additive cocycles for the vertical foliation that are Hölder on vertical leaves and holonomy invariant with respect to the horizontal foliation. Such a cocycle induces a distribution on the vertical separatrix that is invariant under the interval exchange maps determined by horizontal returns to finite subsegments. NEWLINENEWLINENEWLINENEWLINE Let \(T = T_{(\lambda,\pi)}\), \(\lambda \in \Lambda_{m} = ({\mathbf R}^{+})^{m}\), \(\pi \in \sigma_{m}\), be an interval exchange on \(I^{\lambda} = [0, |\lambda|)\). Use \(I^{\lambda}_{j}\) to denote the basic intervals of \(T\), i.e., \(I^{\lambda}_{j}=[\beta_{j-1},\beta_{j})\), \(\beta_{k}(\lambda) = \sum_{1 \leq i \leq k}\lambda_{i}\), \(0 \leq k \leq m\). Also, \(T_{(\lambda,\pi)}\) can be defined in terms of an alternating matrix \(L^{\pi}\): \(T_{(\lambda,\pi)}x = x + (L^{\pi}\lambda)_{i}\), \(x \in I^{\lambda}_{j}\). Introduce a space \({Y}_{|\lambda|}\) of functions on \([0, |\lambda|)\) as NEWLINENEWLINENEWLINE\[NEWLINE {Y}_{|\lambda|} = \big\{F \in BV\big([0, |\lambda|)\big)\;:\; F(x^{+}) = F(x),\; 0 \leq x \leq |\lambda|\big\}. NEWLINE\]NEWLINE NEWLINENEWLINESince \(T\) is piecewise isometric and right continuous, the operator defined by \({T}F := F \circ T\) satisfies \({T} {Y}_{|\lambda|} = {Y}_{|\lambda|}\). Define the weak derivative of \(\varphi \in C_{0}([0, |\lambda|))\) as a linear functional on \({Y}_{|\lambda|}\) :NEWLINENEWLINE NEWLINE\[NEWLINE \Lambda_{\varphi}(F) = - \in…
reviewed by: V. V. Chueshev
description: scientific article; zbMATH DE number 6077504