Invariant distributions for interval exchange transformations (Q2906459)

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\[NEWLINE \Lambda_{\varphi}(F) = - \int_{[0, |\lambda|)}\varphi d F + \varphi(|\lambda|) F(|\lambda|^{-}). NEWLINE\]NEWLINE NEWLINENEWLINELet \(C_{0}([0, |\lambda|])\) be the Banach space of continuous functions \(\varphi\) on \([0, |\lambda|]\) such that \(\varphi(0)=0.\) Define NEWLINE\[NEWLINE{K}(T) = \big\{\varphi \in C_{0}([0, |\lambda|]): \Lambda_{\varphi} \circ {T} = \Lambda_{\varphi}\big\}. NEWLINE\]NEWLINENEWLINENEWLINEThe main result of the paper isNEWLINENEWLINETheorem 1.1. Let \(T = T_{(\lambda,\pi)}\), \(\lambda \in \Lambda_{m}\), \(\pi \in \sigma_{m}\), be an interval exchange on \(I^{\lambda}\), and assume that \(T\) satisfies Keane's conditions. Then the dimension of \({K}(T)\) satisfies the inequality NEWLINE\[NEWLINE \dim\big({K}(T)\big) \leq \frac{1}{2}\text{Rank}(L^{\pi}). NEWLINE\]NEWLINENEWLINENEWLINEFor the entire collection see [Zbl 1237.37004].