On the decay of correlation for piecewise monotonic mappings. II
[For part I see ibid. 8, 389-414 (1985; Zbl 0597.58014).] The author considers a class of mappings F of the unit interval \(I=[0,1]\) into itself which are piecewise linear with nonzero slope on every subinterval of a (finite or countable) partition of I with the additional property that every subinterval (except one fixed) has an image enclosing (0,1). After introducing such technical notions as word (a finite sequence of names of subintervals), admissible word, type of a word, generating functions and Fredholm eigenvalue of F a so-called renewal equation for admissible words is derived. This is the main tool for proving the two theorems of the paper: If the infimum \(\xi\) of the lower Lyapunov numbers is positive and the second Fredholm eigenvalue \(\eta\) of F is less than one then F has a unique invariant distribution \(\mu\) being absolutely continuous with respect to Lebesgue measure. Moreover, the dynamical system ([0,1],F,\(\mu)\) is mixing and the correlations for any pair of functions decay asymptotically by a power law with rate \(\eta^ n\). If \(\xi\) is negative then there exists an attracting periodic orbit.
scientific article; zbMATH DE number 4005199