Computability of Brolin-Lyubich Measure

Abstract

Brolin-Lyubich measure $\lambda_R$ of a rational endomorphism $R:\riem\to\riem$ with $\deg R\geq 2$ is the unique invariant measure of maximal entropy $h_{\lambda_R}=h_{\text{top}}(R)=\log d$. Its support is the Julia set $J(R)$. We demonstrate that $\lambda_R$ is always computable by an algorithm which has access to coefficients of $R$, even when $J(R)$ is not computable. In the case when $R$ is a polynomial, Brolin-Lyubich measure coincides with the harmonic measure of the basin of infinity. We find a sufficient condition for computability of the harmonic measure of a domain, which holds for the basin of infinity of a polynomial mapping, and show that computability may fail for a general domain

Probability, statistics and computation in dynamical systems

“…The proof of this fact is not very difficult, but it does require us to overcome some problems, such as the fact that the function μ → μ(A) is not computable -for more details, see (Binder et al (2011)). …”

“…Binder et al 2011). The Brolin-Lyubich measure of computable quadratic polynomials is always computable.…”

Carathéodory Convergence and Harmonic Measure

“…The harmonic measure of a domain can be approximated by a weakly converging sequence of finitely supported measures, which can be computed given an approximation of the domain (cf. [1]); and so on.…”

Tight space-noise tradeoffs in computing the ergodic measure

“…Ideally, given a dynamical system, we would like to be able to decide properties of its asymptotic behavior or to compute (to within some approximation) the invariant objects describing it. Unfortunately, in many cases, simple questions regarding this behavior are undecidable [Moo90,AMP95,Wol02,KL09] and computing the relevant invariant objects is impossible [BY06,BY07,GHR11,BBRM11]. The general phenomenon behind these results is that, for many classes of dynamical systems, it is possible to 'embed' a Turing machine M in the dynamical system so that achieving the algorithmic task we are concerned with is equivalent to deciding whether M halts.…”