Realizing subexponential entropy growth rates by cutting and stacking

Frank  Blume

doi:10.3934/dcdsb.2015.20.3435

Article Contents

2015, Volume 20, Issue 10: 3435-3459. Doi: 10.3934/dcdsb.2015.20.3435

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Realizing subexponential entropy growth rates by cutting and stacking

Frank Blume^1,

1.
Department of Mathematics, John Brown University, 2000 W. University St, Siloam Springs, AR 72761

Received: September 30, 2014

Revised: February 28, 2015

Published: November 2015

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Abstract

We show that for any concave positive function $f$ defined on $[0,\infty)$ with $\lim_{x\rightarrow\infty}f(x)/x=0$ there exists a rank one system $(X_f,T_f)$ such that $\limsup_{n\rightarrow\infty} H(\alpha_0^{n-1})/f(n)\ge 1$ for all nontrivial partitions $\alpha$ of $X_f$ into two sets and that there is one partition $\alpha$ of $X_f$ into two sets for which the limit superior of $H(\alpha_0^{n-1})/f(n)$ is equal to one whenever the condition $\lim_{x\rightarrow\infty}\ln x/f(x)=0$ is satisfied. Furthermore, for each system $(X_f,T_f)$ we also identify the minimal entropy growth rate in the limit inferior.

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Mathematics Subject Classification: Primary: 28D20; Secondary: 27A99.

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References

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