We propose a notion of Frobenius-Perron dimension for certain free Z \mathbb {Z} </mml:semant...

The Phillips-Perron unit root test was used to test for unit roots, using level or trend nonstationarity as the null [1]. (A unit root tests whether a variable y evolves as a random walk, i.e., with the dynamic relation yt = yt-1 + constant + deterministic trend + errort.) The KPSS test [1,2], which uses level or trend stationarity as the null, was used to confirm the results of the Phillips-Pe...

We consider S-adic expansions associated with continued fraction algorithms, where an S-adic expansion corresponds to an infinite composition of substitutions. Recall that a substitution is a morphism of the free monoid. We focus in particular on the substitutions associated with regular continued fractions (Sturmian substitutions), and with Arnoux–Rauzy, Brun, and Jacobi–Perron (multidimension...

We prove that for all algebraic number β > 1 the strings of zeros in the Rényi βexpansion dβ(1) of 1 exhibit a lacunarity bounded above by log(s(Pβ))/ log(β), where s(Pβ) is the size of the minimal polynomial of β. The conjecture about the specification of the β-shift, equivalently the uniform discreteness of the sets Zβ of β-integers, for β a Perron number is discussed. We propose a classifica...

We prove that the Frobenius–Perron operator of the cusp map F : [−1, 1] → [−1, 1], F (x) = 1 − 2 √ |x| (which is an approximation of the Poincaré section of the Lorenz attractor) has no analytic eigenfunctions corresponding to eigenvalues different from 0 and 1. We also prove that the spectrum of the Frobenius–Perron operator in the space of real-analytic functions on [0, 1] as well as in the s...

By a nonnegative matrix we mean a matrix whose entries are nonnegative real numbers. By positive matrix we mean a matrix all of whose entries are strictly positive real numbers. These notes give the core elements of the Perron-Frobenius theory of nonnegative matrices. This splits into three parts: (1) the primitive case (due to Perron) (2) the irreducible case (due to Frobenius) (3) the general...

We compare computational cost and accuracy between two different approaches of dispersion analysis. One is the conventional Monte Carlo method and the other being the Perron-Frobenius operator approach, that directly propagates the joint probability density function using Liouville equation. It is shown that with same computational budget, Perron-Frobenius operator approach rewards better accur...