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 and Continuous Dynamical Systems, 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. 

[2]

N. Aoki and K. Hiraide, Topological Theory of Dynamical Systems, North-Holland Publishing Co. , Amsterdam, 1994.

[3]

A. Artigue, Robustly N-expansive surface diffeomorphisms, Discrete Contin. Dyn. Syst, 36 (2016), 2367-2376.  doi: 10.3934/dcds.2016.36.2367.

[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.

[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.

[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.

[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.

[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.

[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.

[10]

A. CastroK. 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.

[11]

M. Denker, S. Senti and X. Zhang, Fluctuations of ergodic sums over periodic orbits under specification, Preprint.

[12]

T. EirolaO. Nevanlinna and S. Pilyugin, Limit shadowing property, Numer. Funct. Anal. Optim., 18 (1997), 75-92.  doi: 10.1080/01630569708816748.

[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.

[14]

H. Kato, Continuum-wise expansive homeomorphisms, Canad. J. Math., 45 (1993), 576-598.  doi: 10.4153/CJM-1993-030-4.

[15]

C. A. Morales, A generalization of expansivity, Discrete Contin. Dyn. Syst, 32 (2012), 293-301.  doi: 10.3934/dcds.2012.32.293.

[16]

C. A. Morales and V. F. Sirvent, Expansive Measures 29 Colóquio Brasileiro de Matemática, 2013.

[17]

A. V. OsipovS. Y. Pilyugin and S. Tikhomirov, Periodic shadowing and $Ω$-stability, Regular and Chaotic Dynamics, 15 (2010), 404-417.  doi: 10.1134/S1560354710020255.

[18]

S. Y. Pilyugin, Shadowing in structurally stable flows, Journal of differential equations, 140 (1997), 238-265.  doi: 10.1006/jdeq.1997.3295.

[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.

[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.

[21]

S. Smale, Dynamical systems on n-dimensional manifolds, in Symposium on differential equations and dynamical systems (Puerto Rico) Academic Press, New York, 1967.

[22]

W. R. Utz, Unstable homeomorphisms, Proc. Amer. Math. Soc., 1 (1950), 769-774.  doi: 10.1090/S0002-9939-1950-0038022-3.

show all references

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. 

[2]

N. Aoki and K. Hiraide, Topological Theory of Dynamical Systems, North-Holland Publishing Co. , Amsterdam, 1994.

[3]

A. Artigue, Robustly N-expansive surface diffeomorphisms, Discrete Contin. Dyn. Syst, 36 (2016), 2367-2376.  doi: 10.3934/dcds.2016.36.2367.

[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.

[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.

[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.

[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.

[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.

[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.

[10]

A. CastroK. 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.

[11]

M. Denker, S. Senti and X. Zhang, Fluctuations of ergodic sums over periodic orbits under specification, Preprint.

[12]

T. EirolaO. Nevanlinna and S. Pilyugin, Limit shadowing property, Numer. Funct. Anal. Optim., 18 (1997), 75-92.  doi: 10.1080/01630569708816748.

[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.

[14]

H. Kato, Continuum-wise expansive homeomorphisms, Canad. J. Math., 45 (1993), 576-598.  doi: 10.4153/CJM-1993-030-4.

[15]

C. A. Morales, A generalization of expansivity, Discrete Contin. Dyn. Syst, 32 (2012), 293-301.  doi: 10.3934/dcds.2012.32.293.

[16]

C. A. Morales and V. F. Sirvent, Expansive Measures 29 Colóquio Brasileiro de Matemática, 2013.

[17]

A. V. OsipovS. Y. Pilyugin and S. Tikhomirov, Periodic shadowing and $Ω$-stability, Regular and Chaotic Dynamics, 15 (2010), 404-417.  doi: 10.1134/S1560354710020255.

[18]

S. Y. Pilyugin, Shadowing in structurally stable flows, Journal of differential equations, 140 (1997), 238-265.  doi: 10.1006/jdeq.1997.3295.

[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.

[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.

[21]

S. Smale, Dynamical systems on n-dimensional manifolds, in Symposium on differential equations and dynamical systems (Puerto Rico) Academic Press, New York, 1967.

[22]

W. R. Utz, Unstable homeomorphisms, Proc. Amer. Math. Soc., 1 (1950), 769-774.  doi: 10.1090/S0002-9939-1950-0038022-3.

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