Compactness of eigenvectors for certain families of non-negative matrices (Q805709)

Let \(\tau\) : [0,1]\(\to [0,1]\) be a piecewise expanding map which is Markov with respect to a partition \({\mathcal I}\) (i.e. \(\tau\) transforms the set of end-points of intervals of \({\mathcal I}\) into itself). Let \({\mathcal C}\) be the class of Markov transformations which are piecewise linear on some partition \({\mathcal I}=\{[a_ i,a_{i+1}]\}^{N- 1}_{i=0}\). If \(\tau\in {\mathcal C}\) then \(\tau\) induces an \(N\times N\) matrix \(M_{\tau}=[m_{jk}]\) where \(m_{jk}=(1/| \tau_ j'|)\delta_{jk}\) where \(\tau_ j'=(d\tau /dx)|_{x\in I_ j}\), \(\delta_{jk}=1\) if \(I_ k\subset \tau (I_ j)\) and 0 otherwise (where \(I_ k=[a_ k,a_{k+1}])\). It is known that \(M_{\tau}\) is the Frobenius-Perron operator \(P_{\tau}\), restricted to the piecewise constant functions on the partition \({\mathcal I}\) defining \(\tau\) [cf. the first author and \textit{M. Scarowsky}, Trans. Am. Math. Soc. 255, 243-262 (1979; Zbl 0418.28010)]. Also \(M_{\tau}\) is similar to a stochastic matrix [cf. \textit{N. Friedman} and the first author, Linear Algebra Appl. 38, 141-147 (1981; Zbl 0469.15008)]. Sufficient conditions are given on a class of nonnegative matrices \({\mathcal M}\) so that the normalized left eigenvectors of the matrices in \({\mathcal M}\), interpreted as functions on the unit interval, form a precompact set in \(L_ 1\). These results have important consequences in the study of limiting properties of Markov chains.