We consider small random perturbations of a large class of nonuniformly hy-perbolic unimodal maps and prove stochastic stability in the strong sense (L 1-convergence of invariant densities) and uniform bounds for the exponential rate of decay of correlations. Our method is based on an analysis of the spectrum of a modiied Perron-Frobenius operator for a tower extension of the Markov chain.

Consider a compact metric space $$(M, d_M)$$ and $$X = M^{{\mathbb {N}}}$$ . We prove Ruelle’s Perron Frobenius Theorem for class of subshifts with Markovian structure introduced in da Silva et al. (Bull Braz Math Soc 45:53–72, 2014) which are defined from continuous function $$A : M \times \rightarrow {\mathbb {R}}$$ that determines the set admissible sequences. In particular, this includes fi...

We consider small random perturbations of expanding and piecewise expanding maps and prove the robustness of their invariant densities and rates of mixing. We do this by proving some simple lemmas about the robustness of the spectra of certain operators. These abstract results are then applied to the Perron-Frobenius operators of the models in question. Introduction Let f : M → M be a dynamical...

Eigenvalue spectrum of adjacency matrices of many complex networks reveals that a large real eigenvalue separate from the bulk of the population of eigenvalues. A related theorem in this observation is that of Perron-Frobenius, which states that ifˆA is a positive matrix, then there exists a unique eigenvalue ofˆA, which has the greatest absolute value, and its associated eigenvector may be tak...

Abstract. In this article we prove a local large deviation principle (LLDP) for the critical multitype Galton-Watson process from spectral potential point. We define the so-called a spectral potential UK( ·, π) for the Galton-Watson process, where π is the normalized eigen vector corresponding to the leading Perron-Frobenius eigen value 1l of the transition matrix A(·, ·) defined from K, the tr...

The generalized spectral decomposition of the Frobenius-Perron operator of the tent map with varying height is determined at the band-splitting points. The decomposition includes both decay onto the attracting set and the approach to the asymptotically periodic state on the attractor. Explicit compact expressions for the polynomial eigenstates are obtained using algebraic techniques. (c) 1998 A...

We use a recently found parametrization of the solutions of the inverse Frobenius-Perron problem within the class of complete unimodal maps to develop a Monte-Carlo approach for the construction of one-dimensional chaotic dynamical laws with given statistical properties, i.e. invariant density and autocorrelation function. A variety of different examples are presented to demonstrate the power o...

We will give a summary about the relations between the spectra of the Perron–Frobenius operator and pseudo random sequences for 1-dimensional cases. There are many difficulties to construct general theory of higher-dimensional cases. We will give several examples for these cases.

We describe a computational method of approximating the \physical" or Sinai-Bowen-Ruelle measure of an Anosov system in two dimensions. The approximation may either be viewed as a xed point of an approximate Perron-Frobenius operator or as an invariant measure of a randomly perturbed system.