# American Institute of Mathematical Sciences

April  2017, 37(4): 1941-1957. doi: 10.3934/dcds.2017082

## On specification and measure expansiveness

 1 Department of Mathematics, Universidade Federal do Rio de Janeiro, Rio de Janeiro, CEP 21941-909 RJ, Brazil 2 Department of Mathematics, The Pennsylvania State University, University Park, PA 16802, USA

Received  June 2016 Revised  November 2016 Published  December 2016

Fund Project: The research of W.C. was supported by CAPES and CNPq, and of M.D. by Johann-Gottfried-Herder Foundation

We relate the local specification and periodic shadowing properties. We also clarify the relation between local weak specification and local specification if the system is measure expansive. The notion of strong measure expansiveness is introduced, and an example of a non-expansive systems with the strong measure expansive property is given. Moreover, we find a family of examples with the $N$-expansive property, which are not strong measure expansive. We finally show a spectral decomposition theorem for strong measure expansive dynamical systems with shadowing.

Citation: Welington Cordeiro, Manfred Denker, Xuan Zhang. On specification and measure expansiveness. Discrete & Continuous Dynamical Systems - A, 2017, 37 (4) : 1941-1957. doi: 10.3934/dcds.2017082
##### References:
 [1] D. V. Anosov, On a class of invariant sets of smooth dynamical systems, Proc. 5th Int. Conf. on Nonlin. Oscill., 2 (1970), 39-45. Google Scholar [2] N. Aoki and K. Hiraide, Topological Theory of Dynamical Systems, North-Holland Publishing Co. , Amsterdam, 1994. Google Scholar [3] A. Artigue, Robustly N-expansive surface diffeomorphisms, Discrete Contin. Dyn. Syst, 36 (2016), 2367-2376. doi: 10.3934/dcds.2016.36.2367. Google Scholar [4] A. Artigue and D. Carrasco-Olivera, A note on measure-expansive diffeomorphisms, J. Math. Anal. Appl., 428 (2015), 713-716. doi: 10.1016/j.jmaa.2015.02.052. Google Scholar [5] A. Artigue, M. J. Pacífico and J. Vieitez, N-expansive homeomorphisms on surfaces Communications in Contemporary Mathematics 19 (2017), 1650040, 18pp. doi: 10.1142/S0219199716500401. Google Scholar [6] R. Bowen, Periodic points and measures for Axiom A diffeomorphisms, Trans. Amer. Math. Soc., 154 (1971), 377-397. doi: 10.1090/S0002-9947-1971-0282372-0. Google Scholar [7] 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 [8] B. Carvalho and W. Cordeiro, N-expansive homeomorphisms with the shadowing property, Journal of Differential Equations, 261 (2016), 3734-3755. doi: 10.1016/j.jde.2016.06.003. Google Scholar [9] B. Carvalho and D. Kwietniak, On homeomorphisms with the two-sided limit shadowing property, J. Math Anal. Appl., 420 (2014), 801-813. doi: 10.1016/j.jmaa.2014.06.011. Google Scholar [10] A. Castro, K. Oliveira and V. Pinheiro, Shadowing by non-uniformly hyperbolic periodic points and uniform hyperbolicity, Nonlinearity, 20 (2007), 75-85. doi: 10.1088/0951-7715/20/1/005. Google Scholar [11] M. Denker, S. Senti and X. Zhang, Fluctuations of ergodic sums over periodic orbits under specification, Preprint.Google Scholar [12] T. Eirola, O. Nevanlinna and S. Pilyugin, Limit shadowing property, Numer. Funct. Anal. Optim., 18 (1997), 75-92. doi: 10.1080/01630569708816748. Google Scholar [13] P. Kościelniak, On genericity of shadowing and periodic shadowing property, J. Math. Anal. Appl., 310 (2005), 188-196. doi: 10.1016/j.jmaa.2005.01.053. Google Scholar [14] H. Kato, Continuum-wise expansive homeomorphisms, Canad. J. Math., 45 (1993), 576-598. doi: 10.4153/CJM-1993-030-4. Google Scholar [15] C. A. Morales, A generalization of expansivity, Discrete Contin. Dyn. Syst, 32 (2012), 293-301. doi: 10.3934/dcds.2012.32.293. Google Scholar [16] C. A. Morales and V. F. Sirvent, Expansive Measures 29 Colóquio Brasileiro de Matemática, 2013. Google Scholar [17] A. V. Osipov, S. Y. Pilyugin and S. Tikhomirov, Periodic shadowing and $Ω$-stability, Regular and Chaotic Dynamics, 15 (2010), 404-417. doi: 10.1134/S1560354710020255. Google Scholar [18] S. Y. Pilyugin, Shadowing in structurally stable flows, Journal of differential equations, 140 (1997), 238-265. doi: 10.1006/jdeq.1997.3295. Google Scholar [19] S. Y. Pilyugin and S. Tikhomirov, Lipschitz shadowing implies structural stability, Nonlinearity, 23 (2010), 2509-2515. doi: 10.1088/0951-7715/23/10/009. Google Scholar [20] K. Sakai, Various shadowing properties for positively expansive maps, Topology and its Applications, 131 (2003), 15-31. doi: 10.1016/S0166-8641(02)00260-2. Google Scholar [21] S. Smale, Dynamical systems on n-dimensional manifolds, in Symposium on differential equations and dynamical systems (Puerto Rico) Academic Press, New York, 1967. Google Scholar [22] W. R. Utz, Unstable homeomorphisms, Proc. Amer. Math. Soc., 1 (1950), 769-774. doi: 10.1090/S0002-9939-1950-0038022-3. Google Scholar

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##### References:
 [1] D. V. Anosov, On a class of invariant sets of smooth dynamical systems, Proc. 5th Int. Conf. on Nonlin. Oscill., 2 (1970), 39-45. Google Scholar [2] N. Aoki and K. Hiraide, Topological Theory of Dynamical Systems, North-Holland Publishing Co. , Amsterdam, 1994. Google Scholar [3] A. Artigue, Robustly N-expansive surface diffeomorphisms, Discrete Contin. Dyn. Syst, 36 (2016), 2367-2376. doi: 10.3934/dcds.2016.36.2367. Google Scholar [4] A. Artigue and D. Carrasco-Olivera, A note on measure-expansive diffeomorphisms, J. Math. Anal. Appl., 428 (2015), 713-716. doi: 10.1016/j.jmaa.2015.02.052. Google Scholar [5] A. Artigue, M. J. Pacífico and J. Vieitez, N-expansive homeomorphisms on surfaces Communications in Contemporary Mathematics 19 (2017), 1650040, 18pp. doi: 10.1142/S0219199716500401. Google Scholar [6] R. Bowen, Periodic points and measures for Axiom A diffeomorphisms, Trans. Amer. Math. Soc., 154 (1971), 377-397. doi: 10.1090/S0002-9947-1971-0282372-0. Google Scholar [7] 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 [8] B. Carvalho and W. Cordeiro, N-expansive homeomorphisms with the shadowing property, Journal of Differential Equations, 261 (2016), 3734-3755. doi: 10.1016/j.jde.2016.06.003. Google Scholar [9] B. Carvalho and D. Kwietniak, On homeomorphisms with the two-sided limit shadowing property, J. Math Anal. Appl., 420 (2014), 801-813. doi: 10.1016/j.jmaa.2014.06.011. Google Scholar [10] A. Castro, K. Oliveira and V. Pinheiro, Shadowing by non-uniformly hyperbolic periodic points and uniform hyperbolicity, Nonlinearity, 20 (2007), 75-85. doi: 10.1088/0951-7715/20/1/005. Google Scholar [11] M. Denker, S. Senti and X. Zhang, Fluctuations of ergodic sums over periodic orbits under specification, Preprint.Google Scholar [12] T. Eirola, O. Nevanlinna and S. Pilyugin, Limit shadowing property, Numer. Funct. Anal. Optim., 18 (1997), 75-92. doi: 10.1080/01630569708816748. Google Scholar [13] P. Kościelniak, On genericity of shadowing and periodic shadowing property, J. Math. Anal. Appl., 310 (2005), 188-196. doi: 10.1016/j.jmaa.2005.01.053. Google Scholar [14] H. Kato, Continuum-wise expansive homeomorphisms, Canad. J. Math., 45 (1993), 576-598. doi: 10.4153/CJM-1993-030-4. Google Scholar [15] C. A. Morales, A generalization of expansivity, Discrete Contin. Dyn. Syst, 32 (2012), 293-301. doi: 10.3934/dcds.2012.32.293. Google Scholar [16] C. A. Morales and V. F. Sirvent, Expansive Measures 29 Colóquio Brasileiro de Matemática, 2013. Google Scholar [17] A. V. Osipov, S. Y. Pilyugin and S. Tikhomirov, Periodic shadowing and $Ω$-stability, Regular and Chaotic Dynamics, 15 (2010), 404-417. doi: 10.1134/S1560354710020255. Google Scholar [18] S. Y. Pilyugin, Shadowing in structurally stable flows, Journal of differential equations, 140 (1997), 238-265. doi: 10.1006/jdeq.1997.3295. Google Scholar [19] S. Y. Pilyugin and S. Tikhomirov, Lipschitz shadowing implies structural stability, Nonlinearity, 23 (2010), 2509-2515. doi: 10.1088/0951-7715/23/10/009. Google Scholar [20] K. Sakai, Various shadowing properties for positively expansive maps, Topology and its Applications, 131 (2003), 15-31. doi: 10.1016/S0166-8641(02)00260-2. Google Scholar [21] S. Smale, Dynamical systems on n-dimensional manifolds, in Symposium on differential equations and dynamical systems (Puerto Rico) Academic Press, New York, 1967. Google Scholar [22] W. R. Utz, Unstable homeomorphisms, Proc. Amer. Math. Soc., 1 (1950), 769-774. doi: 10.1090/S0002-9939-1950-0038022-3. Google Scholar
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