Rényi entropy and recurrence

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June 2013, 33(6): 2403-2421. doi: 10.3934/dcds.2013.33.2403

Rényi entropy and recurrence

1.	Mathematics Department, USC, Los Angeles, CA 90089-1113, United States

Received February 2012 Revised July 2012 Published December 2012

Abstract
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This paper studies the relationship between the return time $\tau_n$ and the Rényi Entropy Function of order $s$, $R(s)$. For a dynamical system with an invariant $\alpha$-mixing measure $\mu$ and a measurable partition, we consider the sum $W$ of measures of cylinders along orbit segments of length $\tau_n$ and relate that growth/decay rate to the R$\acute{\textrm{e}}$nyi Entropy. The key strategy is to introduce the hitting number $\nu_x(A) = | \{1 \leq i \leq \tau_n(x) : T^i(x) \in A\}|$, the number of times that $x$ hits the set $A$ when $x$ travels along its orbit of length $\tau_n(x)$, and write $W=\sum \nu_x(A) \mu(A)^s$, where the sum is taken over the $n$-cylinders. Then we show that $\nu_x(A) \approx \exp(n h_{\mu}) \mu(A)$ for most $n$-cylinders $A$. Hence $W \approx \exp(nh_{\mu}) \sum \mu(A)^{1+s}$, which relates $\tau_n(x)$ to $R(s)$, as the sum $\sum \mu(A)^{1+s} \approx \exp(-nsR(s))$.

Keywords: $\alpha$-mixing, countably infinite partition., return time, Rényi entropy.

Mathematics Subject Classification: Primary: 37A50, 28D05; Secondary: 60E1.

Citation: Milton Ko. Rényi entropy and recurrence. Discrete and Continuous Dynamical Systems, 2013, 33 (6) : 2403-2421. doi: 10.3934/dcds.2013.33.2403

References:

[1]	K. Agyem, J. M. Arbeit, R. W. Fuhrhop, M. S. Hughes, G. M. Lanza, J. E. McCarthy, J. N. Marsh, R. G. Neumann, J. Smith, T. Thomas, K. D. Wallace and S. A. Wickline, Application of rényi entropy for Ultrasonic molecular imaging, Journal of the Acoustical Society of Americal, 125 (2009), 3141-3145.
[2]	K. Agyem, J. M. Arbeit, R. W. Fuhrhop, G. M. Lanza, J. E. McCarthy, J. N. Marsh, R. G. Neumann, J. Smith, T. Thomas, K. D. Wallace and S. A. Wickline, "Application of Rényi Entropy to Detect Subtle Changes in Scattering Architecture,", 2008. Available from: , ().
[3]	R. Bowen, "Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms,", Springer Lecture Notes in Mathematics 470., ().
[4]	V. M. Deschamps, B. Schmitt, M. Urbanski and A. Zdunik, Pressure and recurrence, Fund. Math., 178 (2003), 129-141. doi: 10.4064/fm178-2-3.
[5]	N. Haydn and S Vaienti, The rényi entropy function and the large deviation of short return times, Ergodic Theory and Dynamical System, 39 (2010), 159-179. doi: 10.1017/S0143385709000030.
[6]	J. Baez, "Rényi Entropy and Free Energy,", 2011. Available from: , ().
[7]	R. Mañé, "Ergodic Theory and Differential Dynamics," Springer, 1985.
[8]	D. Ornstein and B. Weiss, Entropy and data compression schemes, IEEE Trans. Inf. Theory, 39 (1993), 78-83. doi: 10.1109/18.179344.
[9]	D. Ornstein and B. Weiss, Entropy and recurrence rates for stationary random fields, IEEE Trans. Inf. Theory, 48 (1993), 1694-1697. doi: 10.1109/TIT.2002.1003848.
[10]	A. Rényi, On measures of entropy and information, Proc. Fourth Berkeley Symp. on Math. Statist. and Prob., 1 (1961), 547-561.
[11]	F. Takens and E. Verbitsky, Generalised entropies, rényi and correlation integral approach, Nonlinearity, 4 (1998), 771-782. doi: 10.1088/0951-7715/11/4/001.

show all references

References:

[1]	K. Agyem, J. M. Arbeit, R. W. Fuhrhop, M. S. Hughes, G. M. Lanza, J. E. McCarthy, J. N. Marsh, R. G. Neumann, J. Smith, T. Thomas, K. D. Wallace and S. A. Wickline, Application of rényi entropy for Ultrasonic molecular imaging, Journal of the Acoustical Society of Americal, 125 (2009), 3141-3145.
[2]	K. Agyem, J. M. Arbeit, R. W. Fuhrhop, G. M. Lanza, J. E. McCarthy, J. N. Marsh, R. G. Neumann, J. Smith, T. Thomas, K. D. Wallace and S. A. Wickline, "Application of Rényi Entropy to Detect Subtle Changes in Scattering Architecture,", 2008. Available from: , ().
[3]	R. Bowen, "Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms,", Springer Lecture Notes in Mathematics 470., ().
[4]	V. M. Deschamps, B. Schmitt, M. Urbanski and A. Zdunik, Pressure and recurrence, Fund. Math., 178 (2003), 129-141. doi: 10.4064/fm178-2-3.
[5]	N. Haydn and S Vaienti, The rényi entropy function and the large deviation of short return times, Ergodic Theory and Dynamical System, 39 (2010), 159-179. doi: 10.1017/S0143385709000030.
[6]	J. Baez, "Rényi Entropy and Free Energy,", 2011. Available from: , ().
[7]	R. Mañé, "Ergodic Theory and Differential Dynamics," Springer, 1985.
[8]	D. Ornstein and B. Weiss, Entropy and data compression schemes, IEEE Trans. Inf. Theory, 39 (1993), 78-83. doi: 10.1109/18.179344.
[9]	D. Ornstein and B. Weiss, Entropy and recurrence rates for stationary random fields, IEEE Trans. Inf. Theory, 48 (1993), 1694-1697. doi: 10.1109/TIT.2002.1003848.
[10]	A. Rényi, On measures of entropy and information, Proc. Fourth Berkeley Symp. on Math. Statist. and Prob., 1 (1961), 547-561.
[11]	F. Takens and E. Verbitsky, Generalised entropies, rényi and correlation integral approach, Nonlinearity, 4 (1998), 771-782. doi: 10.1088/0951-7715/11/4/001.

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