Abraham Boyarsky

Absolutely continuous invariant measures for piecewise expandingC 2 transformations inR N

LetS be a bounded region inRN and let ℊ={Si}i=1/m be a partition ofS into a finite number of subsets having piecewiseC2 boundaries. We assume that whereC2 segments of the boundaries meet, the angle subtended by tangents to these segments at the point of contact is bounded away from 0. Letτ:S →S be piecewiseC2 on ℊ and expanding in the sense that there exists 0<σ< 1 such that for anyi=1, 2, ...,m, ‖Dτi−1‖<σ, whereDτi−1 is the derivative matrix ofτi−1 and ‖ ‖ is the euclidean matrix norm. The main result provides an upper bound onσ which guarantees the existence of an absolutely continuous invariant measure forτ.

On a class of transformations which have unique absolutely continuous invariant measures

A class of piecewise C2 transformations from an interval into itself with slopes greater than 1 in absolute value, and having the property that it takes partition points into partition points is shown to have unique absolutely continuous invariant measures. For this class of functions, a central limit theorem holds for all real measurable functions. For the subclass of piecewise linear transformations having a fixed point, it is shown that the unique absolutely continuous invariant measures are piecewise constant.

Absolutely Continuous Invariant Measures for Piecewise Expanding C 2 Transformations in R N

Let S be a bounded region in R N and let \(P = \left\{ {{S_I}} \right\}_{i - 1}^m\) be a partition of S into a finite number of subsets having piecewise C 2 boundaries. We assume that where C 2 segments of the boundaries meet, the angle subtended by tangents to these segments at the point of contact is bounded away from 0. Let τ: S → S be piecewise C 2 on P and expanding in the sense that there exists 0 < σ < 1 such that for any i = 1, 2,…,m, \( ||D{\tau _i}^{ - 1}||{\text{ < }}\sigma \) where Dτ i -1 is the derivative matrix of τ i -1 and || || is the euclidean matrix norm. The main result provides an upper bound on σ which guarantees the existence of an absolutely continuous invariant measure for τ.

Why computers like lebesgue measure

Let τ: [0,1]→[0,1] be a transformation which has an absolutely continuous invariant measure μ. Let τ be a realistic, deterministic model for τ. We prove that if τ has long computer trajectories, either periodic or non-periodic, then these computer trajectories have histograms which approach the density of μ. For a large class of piecewise linear transformations, we prove the existence of long periodic trajectories.

Randomness implies order

Abstract Let τ: [0, 1] → [0, 1] possess a unique invariant density f ∗ . Then given any ϵ > 0, we can find a density function p such that ∥ p − f ∗ ∥ and p is the invariant density of the stochastic difference equation xn + 1 = τ(xn) + W, where W is a random variable. It follows that for all starting points x 0 ϵ [0, 1], lim n→∞ (1 n) ∑ i = 0 n − 1 χ B (x i ) = ∝ B p(ξ) dξ .

A dynamical system model for interference effects and the two-slit experiment of quantum physics

Abstract Given two probability density functions ƒ 1 and ƒ 2 , a method is described for combining the densities in a physically meaningful way. The method involves the construction of underlying transformations τ 1 and τ 2 , and then forming a dynamical system from these two transformations referred to as a random transformation. The invariant (stationary) probability density function for the random transformation is the “combined” density of ƒ 1 and ƒ 2 . This method of combining probability density functions is used to model interference effects in physical systems. In particular, the dynamics of the two-slit experiment of quantum physics is modelled by an appropriate random transformation. Computer results are presented which qualitatively appear like experimental results. The notion of wave is not needed in the model.

Computing the topological entropy of general one-dimensional maps

A matrix-theoretic method for computing the topological entropy of continuous, piecewise monotonic maps of the interval is presented. The method results in a constructive procedure which is easily implemented on the computer. Examples for families of unimodal, nonunimodal and discontinuous maps are presented.

A matrix solution to the inverse Perron-Frobenius problem

Let f be a probability density function on the unit interval I. The inverse Perron-Frobenius problem involves determining a transformation τ: I → I such that the one-dimensional dynamical system x i+1 = τ(x i ) has f as its unique invariant density function. A matrix method is developed that provides a simple relationship between τ and f, where f is any piecewise constant density function. The result is useful for modelling and predicting chaotic data

Matrices and eigenfunctions induced by Markov maps

Abstract The matrix M induced by a Markov map has 1 as the eigenvalue of maximum modulus. If M is also irreducible, then the algebraic and geometric multiplicities of the eigenvalue 1 are also 1. Let g =( g 1 , g 2 … g n ) be an n -row-vector such that g 1 = c > and g i c , i ⩾2. Then every such g is the unique (upto constant multiples) left eigenvector associated with the eigenvalue 1 of a matrix M induced by a Markov map.

All invariant densities of piecewise linear Markov maps are piecewise constant

Let @t: [0,1] -> [0,1] be a piecewise linear Markov map. It is shown that all density functions invariant under @t must be piecewise constant. This has useful application to solutions of functional equations.