Invariant measures of dynamical systems play important role in understanding the chaotic nature of dynamical systems. Let (I,B, λ) be a normalized measure space, where I = [0, 1], B is a Borel σ-algebra of subsets of I , λ Lebesgue measure in (I,B). Let τ : (I,B, λ) → (I,B, λ) be a deterministic dynamical system. The Frobenius-Perron operator Pτ of τ is a linear operator Pτ : L (I,B)→ L(I,B) de...

I illustrate a uniied approach to the study of the decay of correlations in hyperbolic dynamical systems. The decay of correlations or, alternatively, the rate of approach of some initial distribution to an invariant one, is a widely studied problem in dynamical systems as well as in other elds. Although a full understanding is not yet available, some notable results have been obtained in the c...

Using the Perron-Frobenius operator we establish a new functional central limit theorem result for non-invertible measure preserving maps that are not necessarily ergodic. We apply the result to asymptotically periodic transformations and give an extensive specific example using the tent map.

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.

In this paper we extend work of Kawamura, see [3], for CuntzKrieger algebras OA for infinite matrices A. We generalize the definition of branching systems, prove their existence for any given matrix A and show how they induce some very concrete representations of OA. We use these representations to describe the Perron-Frobenius operator, associated to an nonsingular transformation, as an infini...

We study a family of chaotic maps with limit cases the tent map and the cusp map (the cusp family). We discuss the spectral properties of the corresponding Frobenius–Perron operator in different function spaces including spaces of analytic functions. A numerical study of the eigenvalues and eigenfunctions is performed.

We study the Perron-Frobenius operator P of closed dynamical systems and certain open dynamical systems. We prove that the presence of a large positive eigenvalue ρ of P guarantees the existence of a 2-partition of the phase space for which the escape rates of the open systems defined on the two partition sets are both slower than − log ρ. The open systems with slow escape rates are easily iden...