December  2012, 32(12): 4391-4407. doi: 10.3934/dcds.2012.32.4391

Entropy formulas for dynamical systems with mistakes

1. 

Departamento de Matemática, Universidade Federal da Bahia, Av. Ademar de Barros s/n, 40170-110 Salvador, Brazil, Brazil

2. 

Department of Mathematics, Soochow University, Suzhou 215006, China

Received  June 2011 Revised  February 2012 Published  August 2012

We study the recurrence to mistake dynamical balls, that is, dynamical balls that admit some errors and whose proportion of errors decrease tends to zero with the length of the dynamical ball. We prove, under mild assumptions, that the measure-theoretic entropy coincides with the exponential growth rate of return times to mistake dynamical balls and that minimal return times to mistake dynamical balls grow linearly with respect to its length. Moreover we obtain averaged recurrence formula for subshifts of finite type and suspension semiflows. Applications include $\beta$-transformations, Axiom A flows and suspension semiflows of maps with a mild specification property. In particular we extend some results from [6, 10, 19] for mistake dynamical balls.
Citation: Jérôme Rousseau, Paulo Varandas, Yun Zhao. Entropy formulas for dynamical systems with mistakes. Discrete & Continuous Dynamical Systems - A, 2012, 32 (12) : 4391-4407. doi: 10.3934/dcds.2012.32.4391
References:
[1]

V. Afraimovich, J.-R. Chazottes and B. Saussol, Pointwise dimensions for Poincaré recurrences associated with maps and special flows,, Discrete Contin. Dyn. Syst., 9 (2003), 263.

[2]

L. Barreira and B. Saussol, Multifractal analysis of hyperbolic flows,, Comm. Math. Phys., 214(2) (2000), 339. doi: 10.1007/s002200000268.

[3]

A. M. Blokh, Decomposition of dynamical systems on an interval,, Uspekhi Mat. Nauk, 38(5(233)) (1983), 179.

[4]

R. Bowen, Entropy for group endomorphisms and homogeneous spaces,, Trans. Amer. Math. Soc., 153 (1971), 401. doi: 10.1090/S0002-9947-1971-0274707-X.

[5]

J. Buzzi, Specification on the interval,, Trans. Amer. Math. Soc., 349 (1997), 2737. doi: 10.1090/S0002-9947-97-01873-4.

[6]

J.-R. Chazottes, Poincaré recurrences and entropy of suspended flows,, Comptes Rendus Math., 332 (2001), 739.

[7]

W. Cheng, Y. Zhao and Y. Cao, Pressures for asymptotically sub-additive potentials under a mistake function,, Discrete Cont. Dyn. Syst., 32(2) (2012), 487. doi: 10.3934/dcds.2012.32.487.

[8]

T. Downarowicz and B. Weiss, Entropy theorems along times when $x$ visits a set,, Illinois J. Math., 48 (2004), 59.

[9]

A. Katok, Lyapunov exponents, entropy and periodic points of diffeomorphisms,, Publ. Math. IHES, 51 (1980), 137.

[10]

V. Maume-Deschamps, B. Schmitt, M. Urbański and A. Zdunik, Pressure and recurrence,, Fund. Math., 178 (2003), 129. doi: 10.4064/fm178-2-3.

[11]

A. Mesón and F. Vericat, Poincaré recurrence and topologicalpressure for homeomorphisms with specification,, Far. East J.Dyn. Sys., 7 (2005), 1.

[12]

K. Oliveira and X. Tian., Nonuniform hyperbolicity and nonuniform specification,, Preprint arxiv:1102.1652., ().

[13]

D. Ornstein and B. Weiss, Entropy and data compression schemes,, IEEE Trans. Inform. Theory, 39(1) (1993), 78. doi: 10.1109/18.179344.

[14]

C.-E. Pfister and W.G. Sullivan, On the topological entropy of saturated sets,, Ergod. Th. Dynam. Syst., 27 (2007), 929. doi: 10.1017/S0143385706000824.

[15]

A. Rényi, Representations for real numbers and their ergodic properties,, Acta Math. Acad. Sci. Hung., 8 (1957), 477.

[16]

J. Rousseau, Recurrence rates for observations of flows,, Ergod. Th. Dynam. Syst., (2011).

[17]

B. Saussol, S. Troubetzkoy and S. Vaienti, Recurrence and Lyapunov exponents,, Mosc. Math. J., 3 (2003), 189.

[18]

D. Thompson, Irregular sets, the $\beta$-transformation and thealmost specification property,, Preprint arxiv:0905.0739., ().

[19]

P. Varandas, Entropy and Poincaré recurrence from a geometrical viewpoint,, Nonlinearity, 22 (2009), 2365. doi: 10.1088/0951-7715/22/10/003.

[20]

P. Varandas, Non-uniform specification and large deviations for weak Gibbs measures,, J. Stat. Phys., 146 (2012), 330. doi: 10.1007/s10955-011-0392-7.

[21]

P. Walters, "An Introduction to Ergodic Theory,", Springer Verlag, (1982).

[22]

Y. Zhao and Y. Cao, Measure-theoretic pressure for sub-additive potentials,, Nonlinear Analysis, 70 (2009), 2237.

show all references

References:
[1]

V. Afraimovich, J.-R. Chazottes and B. Saussol, Pointwise dimensions for Poincaré recurrences associated with maps and special flows,, Discrete Contin. Dyn. Syst., 9 (2003), 263.

[2]

L. Barreira and B. Saussol, Multifractal analysis of hyperbolic flows,, Comm. Math. Phys., 214(2) (2000), 339. doi: 10.1007/s002200000268.

[3]

A. M. Blokh, Decomposition of dynamical systems on an interval,, Uspekhi Mat. Nauk, 38(5(233)) (1983), 179.

[4]

R. Bowen, Entropy for group endomorphisms and homogeneous spaces,, Trans. Amer. Math. Soc., 153 (1971), 401. doi: 10.1090/S0002-9947-1971-0274707-X.

[5]

J. Buzzi, Specification on the interval,, Trans. Amer. Math. Soc., 349 (1997), 2737. doi: 10.1090/S0002-9947-97-01873-4.

[6]

J.-R. Chazottes, Poincaré recurrences and entropy of suspended flows,, Comptes Rendus Math., 332 (2001), 739.

[7]

W. Cheng, Y. Zhao and Y. Cao, Pressures for asymptotically sub-additive potentials under a mistake function,, Discrete Cont. Dyn. Syst., 32(2) (2012), 487. doi: 10.3934/dcds.2012.32.487.

[8]

T. Downarowicz and B. Weiss, Entropy theorems along times when $x$ visits a set,, Illinois J. Math., 48 (2004), 59.

[9]

A. Katok, Lyapunov exponents, entropy and periodic points of diffeomorphisms,, Publ. Math. IHES, 51 (1980), 137.

[10]

V. Maume-Deschamps, B. Schmitt, M. Urbański and A. Zdunik, Pressure and recurrence,, Fund. Math., 178 (2003), 129. doi: 10.4064/fm178-2-3.

[11]

A. Mesón and F. Vericat, Poincaré recurrence and topologicalpressure for homeomorphisms with specification,, Far. East J.Dyn. Sys., 7 (2005), 1.

[12]

K. Oliveira and X. Tian., Nonuniform hyperbolicity and nonuniform specification,, Preprint arxiv:1102.1652., ().

[13]

D. Ornstein and B. Weiss, Entropy and data compression schemes,, IEEE Trans. Inform. Theory, 39(1) (1993), 78. doi: 10.1109/18.179344.

[14]

C.-E. Pfister and W.G. Sullivan, On the topological entropy of saturated sets,, Ergod. Th. Dynam. Syst., 27 (2007), 929. doi: 10.1017/S0143385706000824.

[15]

A. Rényi, Representations for real numbers and their ergodic properties,, Acta Math. Acad. Sci. Hung., 8 (1957), 477.

[16]

J. Rousseau, Recurrence rates for observations of flows,, Ergod. Th. Dynam. Syst., (2011).

[17]

B. Saussol, S. Troubetzkoy and S. Vaienti, Recurrence and Lyapunov exponents,, Mosc. Math. J., 3 (2003), 189.

[18]

D. Thompson, Irregular sets, the $\beta$-transformation and thealmost specification property,, Preprint arxiv:0905.0739., ().

[19]

P. Varandas, Entropy and Poincaré recurrence from a geometrical viewpoint,, Nonlinearity, 22 (2009), 2365. doi: 10.1088/0951-7715/22/10/003.

[20]

P. Varandas, Non-uniform specification and large deviations for weak Gibbs measures,, J. Stat. Phys., 146 (2012), 330. doi: 10.1007/s10955-011-0392-7.

[21]

P. Walters, "An Introduction to Ergodic Theory,", Springer Verlag, (1982).

[22]

Y. Zhao and Y. Cao, Measure-theoretic pressure for sub-additive potentials,, Nonlinear Analysis, 70 (2009), 2237.

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