Invariant measures for maps of continued fraction type
The paper deals with the probabilistic density-function of certain maps, which are generalizations of the continued-fraction transformation \(TB\to B\), where \(B\) is the interval \((0,1)\) and \(Tx=(1/x)\), \((\centerdot)\) being the fractional part. The main result of the paper is the following. Let \(B\) be a subspace of the Euclidean \(n\)-space and \[ T(x_ 1,x_ 2,\ldots,x_ n)=(y_ 1,y_ 2,\ldots,y_ n), \] where \(y_ j=(A_{j0}+A_{j1}x_ 1+\cdots+A_{1n}x_ n)/(A_{00}+A_{01}x_ 1+\cdots+A_{0n}x_ n)\) \((1\leq j\leq n)\). Then under certain conditions on the coefficients \(A_{jk}\), a formula of the form \[ u(x_ 1,x_ 2,\ldots,x_ n)=\int_ B(1+\sum_{1\leq j\leq n} f(t_ 1,t_ 2,\ldots,t_ n)^{- (n+1)}dv(t_ 1,t_ 2,\ldots,t_ n) \] is given for the density-function \(v(x_ 1,x_ 2,\ldots,x_ n)\) of the invariant measure of the transformation \(T\).
scientific article; zbMATH DE number 24721
scientific article