Treffer: Computing Pointwise Fractal Dimension by Conditioning in Multivariate Distributions and Time Series: Computing pointwise fractal dimension by conditioning in multivariate distributions and time series

This paper is concerned with the problem of finding pointwise dimensions for invariant and joint distributions of a time series. Starting point is a conditional additivity rule, which under certain Lispschitz conditions allows to calculate the dimension of the multivariate joint distribution of the first \(n\) values of a time series from conditional distributions of the values at fixed times. This approach is used to analyse the behaviour of pointwise dimensions for various stationary stochastic processes, with an emphasis on dynamical systems corrupted by noise. Examples include randomly iterated function systems and missing data models. A question discussed in several examples is whether, as \(n\to\infty\), the dimension values approach a finite limit. This effect is frequently taken as evidence that the underlying system has settled onto a finite-dimensional attractor.