A lower bound for Garsia's entropy for certain Bernoulli convolutions

Details A lower bound for Garsia's entropy for certain Bernoulli convolutions A lower bound for Garsia's entropy for certain Bernoulli convolutions

2008-11-18

arxiv.org/abs/0811.3009v3

LMS. J. Comput. Math. 13 (2010), 130-143

Mathematics

Dynamical Systems

16 pages, 4 figures

Scientific paper

Let $\beta\in(1,2)$ be a Pisot number and let $H_\beta$ denote Garsia's entropy for the Bernoulli convolution associated with $\beta$. Garsia, in 1963 showed that $H_\beta<1$ for any Pisot $\beta$. For the Pisot numbers which satisfy $x^m=x^{m-1}+x^{m-2}+...+x+1$ (with $m\ge2$) Garsia's entropy has been evaluated with high precision by Alexander and Zagier and later improved by Grabner, Kirschenhofer and Tichy, and it proves to be close to 1. No other numerical values for $H_\beta$ are known. In the present paper we show that $H_\beta>0.81$ for all Pisot $\beta$, and improve this lower bound for certain ranges of $\beta$. Our method is computational in nature.

Affiliated with

Also associated with

No associations

Rating

A lower bound for Garsia's entropy for certain Bernoulli convolutions does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with A lower bound for Garsia's entropy for certain Bernoulli convolutions, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and A lower bound for Garsia's entropy for certain Bernoulli convolutions will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-428773