Invariant measures of higher dimensional transformations are hard to calculate. We present new results on the estimation of absolutely continous invariant measures of mixing transformations , including a new method of proof of Ulam's conjecture. The method involves constructing nite matrix approximations to the Perron-Frobenius operator from increasingly ner partitions of the state space X. We ...

For all hyperbolic polynomials we proved in [11] a Lipschitz estimate of Jacobi matrices built by orthogonalizing polynomials with respect to measures in the orbit of classical Perron-Frobenius-Ruelle operators associated to hyperbolic polynomial dynamics (with real Julia set). Here we prove that for all sufficiently hyperbolic polynomials this estimate becomes exponentially better when the dim...

We study the properties of the Google matrix generated by a coarse-grained Perron-Frobenius operator of the Chirikov typical map with dissipation. The finite-size matrix approximant of this operator is constructed by the Ulam method. This method applied to the simple dynamical model generates directed Ulam networks with approximate scale-free scaling and characteristics being in certain feature...

We construct center-stable and center-unstable manifolds, as well as stable and unstable manifolds, for the nonlinear Klein–Gordon equation with a focusing energy subcritical nonlinearity, associated with a family of solitary waves which is generated from any radial stationary solution by the action of all Lorentz transforms and spatial translations. The construction is based on the graph trans...

Abstract In this paper, we introduce a method for computing rigorous local inclusions of solutions Cauchy problems nonlinear heat equations complex time values. The proof is constructive and provides explicit bounds the inclusion solution problem, which rewritten as zero-finding problem on certain Banach space. Using map operator, construct simplified Newton operator show that it has unique fix...

Using path-integral methods, a formula is deduced for the noise-induced escape rate from an attracting fixed point across an unstable fixed point in one-dimensional maps. The calculation starts from the trace formula for the eigenvalues of the Frobenius-Perron operator ruling the time evolution of the probability density in noisy maps. The escape rate is determined from the loop formed by two h...

The inverse Frobenius–Perron problem (IFPP) is a global open-loop strategy to control chaos. The goal of our IFPP is to design a dynamical system in < which is: (1) nearby the original dynamical system, and (2) has a desired invariant density. We reduce the question of stabilizing an arbitrary invariant measure, to the question of a hyperplane intersecting a unit hyperbox; several controllabili...

We study the consequences of deterministic chaos for diffusion-controlled reaction. As an example, we analyze a diffusive-reactive deterministic multibaker and a parameter-dependent variation of it. We construct the diffusive and the reactive modes of the models as eigenstates of the Frobenius-Perron operator. The associated eigenvalues provide the dispersion relations of diffusion and reaction...

We review the basic steps leading to the construction of a Sinai-Ruelle-Bowen (SRB) measure for an innnite lattice of weakly coupled expanding circle maps, and we show that this measure has exponential decay of space-time correlations. First, using the Perron-Frobenius operator, one connects the dynamical system of coupled maps on a d-dimensional lattice to an equilibrium statistical mechanical...

A Markov switching position dependent random map is a random map of a finite number of measurable transformations where the probability of switching from one transformation to another is controlled by a position dependent irreducible stochastic matrix W . Existence of absolutely continuous invariant measures (acim) for a Markov switching position dependent random map was proved in [1] using spe...