American Institute of Mathematical Sciences

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

Realizing subexponential entropy growth rates by cutting and stacking

 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

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.
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.

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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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