Realizing subexponential entropy growth rates by cutting and stacking

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December 2015, 20(10): 3435-3459. doi: 10.3934/dcdsb.2015.20.3435

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, United States

Received October 2014 Revised March 2015 Published September 2015

Abstract
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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.

Keywords: Ergodic theory, entropy growth rate, cutting and stacking, zero-entropy system., rank-one system.

Mathematics Subject Classification: Primary: 28D20; Secondary: 27A9.

Citation: Frank Blume. Realizing subexponential entropy growth rates by cutting and stacking. Discrete and Continuous Dynamical Systems - B, 2015, 20 (10) : 3435-3459. doi: 10.3934/dcdsb.2015.20.3435

References:

[1]	F. Blume, An entropy estimate for infinite interval exchange transformations, Mathematische Zeitschrift, 272 (2012), 17-29. doi: 10.1007/s00209-011-0919-2.
[2]	F. Blume, Minimal rates of entropy convergence for completely ergodic systems, Israel Journal of Mathematics, 108 (1998), 1-12. doi: 10.1007/BF02783038.
[3]	F. Blume, Minimal rates of entropy convergence for rank one systems, Discrete and Continuous Dynamical Systems, 6 (2000), 773-796. doi: 10.3934/dcds.2000.6.773.
[4]	F. Blume, On the relation between entropy and the average complexity of trajectories in dynamical systems, Computational Complexity, 9 (2000), 146-155. doi: 10.1007/PL00001604.
[5]	F. Blume, On the relation between entropy convergence rates and Baire category, Mathematische Zeitschrift, 271 (2012), 723-750. doi: 10.1007/s00209-011-0887-6.
[6]	F. Blume, Possible rates of entropy convergence, Ergodic Theory and Dynamical Systems, 17 (1997), 45-70. doi: 10.1017/S0143385797069733.
[7]	F. Blume, The Rate of Entropy Convergence, Doctoral Dissertation, University of North Carolina at Chapel Hill, 1995.
[8]	A. Katok and J.-P. Thouvenot, Slow entropy type invariants and smooth realization of commuting measure-preserving transformations, Annales de l'Institut Henri Poincare (B) Probability and Statistics, 33 (1997), 323-338. doi: 10.1016/S0246-0203(97)80094-5.
[9]	W. Parry, Entropy and Generators in Ergodic Theory, Benjamin, New York, 1969.
[10]	K. E. Petersen, Ergodic Theory, Cambridge University Press, New York, 1983. doi: 10.1017/CBO9780511608728.

show all references

References:

[1]	F. Blume, An entropy estimate for infinite interval exchange transformations, Mathematische Zeitschrift, 272 (2012), 17-29. doi: 10.1007/s00209-011-0919-2.
[2]	F. Blume, Minimal rates of entropy convergence for completely ergodic systems, Israel Journal of Mathematics, 108 (1998), 1-12. doi: 10.1007/BF02783038.
[3]	F. Blume, Minimal rates of entropy convergence for rank one systems, Discrete and Continuous Dynamical Systems, 6 (2000), 773-796. doi: 10.3934/dcds.2000.6.773.
[4]	F. Blume, On the relation between entropy and the average complexity of trajectories in dynamical systems, Computational Complexity, 9 (2000), 146-155. doi: 10.1007/PL00001604.
[5]	F. Blume, On the relation between entropy convergence rates and Baire category, Mathematische Zeitschrift, 271 (2012), 723-750. doi: 10.1007/s00209-011-0887-6.
[6]	F. Blume, Possible rates of entropy convergence, Ergodic Theory and Dynamical Systems, 17 (1997), 45-70. doi: 10.1017/S0143385797069733.
[7]	F. Blume, The Rate of Entropy Convergence, Doctoral Dissertation, University of North Carolina at Chapel Hill, 1995.
[8]	A. Katok and J.-P. Thouvenot, Slow entropy type invariants and smooth realization of commuting measure-preserving transformations, Annales de l'Institut Henri Poincare (B) Probability and Statistics, 33 (1997), 323-338. doi: 10.1016/S0246-0203(97)80094-5.
[9]	W. Parry, Entropy and Generators in Ergodic Theory, Benjamin, New York, 1969.
[10]	K. E. Petersen, Ergodic Theory, Cambridge University Press, New York, 1983. doi: 10.1017/CBO9780511608728.

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