# American Institute of Mathematical Sciences

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, 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), 263-280.  Google Scholar [2] L. Barreira and B. Saussol, Multifractal analysis of hyperbolic flows, Comm. Math. Phys., 214(2) (2000), 339-371. doi: 10.1007/s002200000268.  Google Scholar [3] A. M. Blokh, Decomposition of dynamical systems on an interval, Uspekhi Mat. Nauk, 38(5(233)) (1983), 179-180.  Google Scholar [4] R. Bowen, Entropy for group endomorphisms and homogeneous spaces, Trans. Amer. Math. Soc., 153 (1971), 401-414. doi: 10.1090/S0002-9947-1971-0274707-X.  Google Scholar [5] J. Buzzi, Specification on the interval, Trans. Amer. Math. Soc., 349 (1997), 2737-2754. doi: 10.1090/S0002-9947-97-01873-4.  Google Scholar [6] J.-R. Chazottes, Poincaré recurrences and entropy of suspended flows, Comptes Rendus Math., 332 (2001), 739-744.  Google Scholar [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), 487-497. doi: 10.3934/dcds.2012.32.487.  Google Scholar [8] T. Downarowicz and B. Weiss, Entropy theorems along times when $x$ visits a set, Illinois J. Math., 48 (2004), 59-69.  Google Scholar [9] A. Katok, Lyapunov exponents, entropy and periodic points of diffeomorphisms, Publ. Math. IHES, 51 (1980), 137-173.  Google Scholar [10] V. Maume-Deschamps, B. Schmitt, M. Urbański and A. Zdunik, Pressure and recurrence, Fund. Math., 178 (2003), 129-141. doi: 10.4064/fm178-2-3.  Google Scholar [11] A. Mesón and F. Vericat, Poincaré recurrence and topologicalpressure for homeomorphisms with specification, Far. East J.Dyn. Sys., 7 (2005), 1-14.  Google Scholar [12] K. Oliveira and X. Tian., Nonuniform hyperbolicity and nonuniform specification,, Preprint arxiv：1102.1652., ().   Google Scholar [13] D. Ornstein and B. Weiss, Entropy and data compression schemes, IEEE Trans. Inform. Theory, 39(1) (1993), 78-83. doi: 10.1109/18.179344.  Google Scholar [14] C.-E. Pfister and W.G. Sullivan, On the topological entropy of saturated sets, Ergod. Th. Dynam. Syst., 27 (2007), 929-956. doi: 10.1017/S0143385706000824.  Google Scholar [15] A. Rényi, Representations for real numbers and their ergodic properties, Acta Math. Acad. Sci. Hung., 8 (1957), 477-493.  Google Scholar [16] J. Rousseau, Recurrence rates for observations of flows, Ergod. Th. Dynam. Syst., Available on CJO 2011. Google Scholar [17] B. Saussol, S. Troubetzkoy and S. Vaienti, Recurrence and Lyapunov exponents, Mosc. Math. J., 3 (2003), 189-203,  Google Scholar [18] D. Thompson, Irregular sets, the $\beta$-transformation and thealmost specification property,, Preprint arxiv：0905.0739., ().   Google Scholar [19] P. Varandas, Entropy and Poincaré recurrence from a geometrical viewpoint, Nonlinearity, 22 (2009), 2365-2375. doi: 10.1088/0951-7715/22/10/003.  Google Scholar [20] P. Varandas, Non-uniform specification and large deviations for weak Gibbs measures, J. Stat. Phys., 146 (2012), 330-358. doi: 10.1007/s10955-011-0392-7.  Google Scholar [21] P. Walters, "An Introduction to Ergodic Theory," Springer Verlag, New York-Berlin, 1982.  Google Scholar [22] Y. Zhao and Y. Cao, Measure-theoretic pressure for sub-additive potentials, Nonlinear Analysis, 70 (2009), 2237-2247.  Google Scholar

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##### 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), 263-280.  Google Scholar [2] L. Barreira and B. Saussol, Multifractal analysis of hyperbolic flows, Comm. Math. Phys., 214(2) (2000), 339-371. doi: 10.1007/s002200000268.  Google Scholar [3] A. M. Blokh, Decomposition of dynamical systems on an interval, Uspekhi Mat. Nauk, 38(5(233)) (1983), 179-180.  Google Scholar [4] R. Bowen, Entropy for group endomorphisms and homogeneous spaces, Trans. Amer. Math. Soc., 153 (1971), 401-414. doi: 10.1090/S0002-9947-1971-0274707-X.  Google Scholar [5] J. Buzzi, Specification on the interval, Trans. Amer. Math. Soc., 349 (1997), 2737-2754. doi: 10.1090/S0002-9947-97-01873-4.  Google Scholar [6] J.-R. Chazottes, Poincaré recurrences and entropy of suspended flows, Comptes Rendus Math., 332 (2001), 739-744.  Google Scholar [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), 487-497. doi: 10.3934/dcds.2012.32.487.  Google Scholar [8] T. Downarowicz and B. Weiss, Entropy theorems along times when $x$ visits a set, Illinois J. Math., 48 (2004), 59-69.  Google Scholar [9] A. Katok, Lyapunov exponents, entropy and periodic points of diffeomorphisms, Publ. Math. IHES, 51 (1980), 137-173.  Google Scholar [10] V. Maume-Deschamps, B. Schmitt, M. Urbański and A. Zdunik, Pressure and recurrence, Fund. Math., 178 (2003), 129-141. doi: 10.4064/fm178-2-3.  Google Scholar [11] A. Mesón and F. Vericat, Poincaré recurrence and topologicalpressure for homeomorphisms with specification, Far. East J.Dyn. Sys., 7 (2005), 1-14.  Google Scholar [12] K. Oliveira and X. Tian., Nonuniform hyperbolicity and nonuniform specification,, Preprint arxiv：1102.1652., ().   Google Scholar [13] D. Ornstein and B. Weiss, Entropy and data compression schemes, IEEE Trans. Inform. Theory, 39(1) (1993), 78-83. doi: 10.1109/18.179344.  Google Scholar [14] C.-E. Pfister and W.G. Sullivan, On the topological entropy of saturated sets, Ergod. Th. Dynam. Syst., 27 (2007), 929-956. doi: 10.1017/S0143385706000824.  Google Scholar [15] A. Rényi, Representations for real numbers and their ergodic properties, Acta Math. Acad. Sci. Hung., 8 (1957), 477-493.  Google Scholar [16] J. Rousseau, Recurrence rates for observations of flows, Ergod. Th. Dynam. Syst., Available on CJO 2011. Google Scholar [17] B. Saussol, S. Troubetzkoy and S. Vaienti, Recurrence and Lyapunov exponents, Mosc. Math. J., 3 (2003), 189-203,  Google Scholar [18] D. Thompson, Irregular sets, the $\beta$-transformation and thealmost specification property,, Preprint arxiv：0905.0739., ().   Google Scholar [19] P. Varandas, Entropy and Poincaré recurrence from a geometrical viewpoint, Nonlinearity, 22 (2009), 2365-2375. doi: 10.1088/0951-7715/22/10/003.  Google Scholar [20] P. Varandas, Non-uniform specification and large deviations for weak Gibbs measures, J. Stat. Phys., 146 (2012), 330-358. doi: 10.1007/s10955-011-0392-7.  Google Scholar [21] P. Walters, "An Introduction to Ergodic Theory," Springer Verlag, New York-Berlin, 1982.  Google Scholar [22] Y. Zhao and Y. Cao, Measure-theoretic pressure for sub-additive potentials, Nonlinear Analysis, 70 (2009), 2237-2247.  Google Scholar
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