July  2019, 39(7): 4041-4056. doi: 10.3934/dcds.2019162

Minimality and gluing orbit property

China Economics and Management Academy, Central University of Finance and Economics, Beijing 100081, China

Received  August 2018 Revised  January 2019 Published  April 2019

Fund Project: The author is supported by National Natural Science Foundation of China, No. 11571387.

We show that a dynamical system with gluing orbit property is either minimal or of positive topological entropy. Moreover, for equicontinuous systems, we show that topological transitivity, minimality and orbit gluing property are equivalent. These facts reflect the similarity and dissimilarity of gluing orbit property with specification like properties.

Citation: Peng Sun. Minimality and gluing orbit property. Discrete & Continuous Dynamical Systems - A, 2019, 39 (7) : 4041-4056. doi: 10.3934/dcds.2019162
References:
[1]

M. BessaM. J. Torres and P. Varandas, On the periodic orbits, shadowing and strong transitivity of continuous flows, Nonlinear Analysis, 175 (2018), 191-209.  doi: 10.1016/j.na.2018.06.002.  Google Scholar

[2]

F. BlanchardE. GlasnerS. Kolyada and A. Maass, On Li-Yorke pairs, J. Reine Angew. Math., 547 (2002), 51-68.  doi: 10.1515/crll.2002.053.  Google Scholar

[3]

T. BomfimM. J. Torres and P. Varandas, Topological features of flows with the reparametrized gluing orbit property, Journal of Differential Equations, 262 (2017), 4292-4313.  doi: 10.1016/j.jde.2017.01.008.  Google Scholar

[4]

T. Bomfim and P. Varandas, The gluing orbit property, uniform hyperbolicity and large deviations principles for semiflows, preprint, arXiv: 1507.03905. Google Scholar

[5]

R. Bowen, Periodic points and measures for Axiom A diffeomorphisms, Trans. Amer. Math. Soc., 154 (1971), 377-397.  doi: 10.2307/1995452.  Google Scholar

[6]

V. Climenhaga and D. J. Thompson, Unique equilibrium states for flows and homeomorphisms with non-uniform structure, Adv. Math., 303 (2016), 745-799.  doi: 10.1016/j.aim.2016.07.029.  Google Scholar

[7]

D. Constantine, J. Lafont and D. Thompson, The weak specification property for geodesic flows on CAT(-1) spaces, preprint, arXiv: 1606.06253. Google Scholar

[8]

M. Denker, C. Grillenberger and K. Sigmund, Ergodic Theory on Compact Spaces, Lecture Notes in Mathematics, Vol. 527, Springer-Verlag, Berlin-New York, 1976.  Google Scholar

[9]

W. Gottschalk and G. Hedlund, Topological Dynamics, Amer. Math. Soc. Colloq. Publ. vol. 36, 1955.  Google Scholar

[10]

F. Hahn and Y. Katznelson, On the entropy of uniquely ergodic transformations, Trans. Amer. Math. Soc., 126 (1967), 335-360.  doi: 10.2307/1994458.  Google Scholar

[11]

M. Herman, Construction d'un difféomorphisme minimal d'entropie topologique non nulle, Ergodic Theory Dynam. Systems, 1 (1981), 65-76.  doi: 10.1017/s0143385700001164.  Google Scholar

[12]

D. KwietniakM. Lacka and P. Oprocha., A panorama of specification-like properties and their consequences, Contemporary Mathematics, 669 (2016), 155-186.  doi: 10.1090/conm/669/13428.  Google Scholar

[13]

M. Misiurewicz and J. Smítal, Smooth chaotic maps with zero topological entropy, Ergodic Theory and Dynamical Systems, 8 (1988), 421-424.  doi: 10.1017/S0143385700004557.  Google Scholar

[14]

W. Sun and X. Tian, Diffeomorphisms with various $C^1$-stable properties, Acta Mathematica Scientia, 32 (2012), 552-558.  doi: 10.1016/S0252-9602(12)60037-X.  Google Scholar

[15]

X. TianS. Wang and X. Wang, Intermediate Lyapunov exponents for system with periodic gluing orbit property, Discrete and Continuous Dynamical Systems - A, 39 (2019), 1019-1032.  doi: 10.3934/dcds.2019042.  Google Scholar

[16]

S. Xiang and Y. Zheng, Multifractal analysis for maps with the gluing orbit property, Taiwanese Journal of Mathematics, 21 (2017), 1099-1113.  doi: 10.11650/tjm/7946.  Google Scholar

show all references

References:
[1]

M. BessaM. J. Torres and P. Varandas, On the periodic orbits, shadowing and strong transitivity of continuous flows, Nonlinear Analysis, 175 (2018), 191-209.  doi: 10.1016/j.na.2018.06.002.  Google Scholar

[2]

F. BlanchardE. GlasnerS. Kolyada and A. Maass, On Li-Yorke pairs, J. Reine Angew. Math., 547 (2002), 51-68.  doi: 10.1515/crll.2002.053.  Google Scholar

[3]

T. BomfimM. J. Torres and P. Varandas, Topological features of flows with the reparametrized gluing orbit property, Journal of Differential Equations, 262 (2017), 4292-4313.  doi: 10.1016/j.jde.2017.01.008.  Google Scholar

[4]

T. Bomfim and P. Varandas, The gluing orbit property, uniform hyperbolicity and large deviations principles for semiflows, preprint, arXiv: 1507.03905. Google Scholar

[5]

R. Bowen, Periodic points and measures for Axiom A diffeomorphisms, Trans. Amer. Math. Soc., 154 (1971), 377-397.  doi: 10.2307/1995452.  Google Scholar

[6]

V. Climenhaga and D. J. Thompson, Unique equilibrium states for flows and homeomorphisms with non-uniform structure, Adv. Math., 303 (2016), 745-799.  doi: 10.1016/j.aim.2016.07.029.  Google Scholar

[7]

D. Constantine, J. Lafont and D. Thompson, The weak specification property for geodesic flows on CAT(-1) spaces, preprint, arXiv: 1606.06253. Google Scholar

[8]

M. Denker, C. Grillenberger and K. Sigmund, Ergodic Theory on Compact Spaces, Lecture Notes in Mathematics, Vol. 527, Springer-Verlag, Berlin-New York, 1976.  Google Scholar

[9]

W. Gottschalk and G. Hedlund, Topological Dynamics, Amer. Math. Soc. Colloq. Publ. vol. 36, 1955.  Google Scholar

[10]

F. Hahn and Y. Katznelson, On the entropy of uniquely ergodic transformations, Trans. Amer. Math. Soc., 126 (1967), 335-360.  doi: 10.2307/1994458.  Google Scholar

[11]

M. Herman, Construction d'un difféomorphisme minimal d'entropie topologique non nulle, Ergodic Theory Dynam. Systems, 1 (1981), 65-76.  doi: 10.1017/s0143385700001164.  Google Scholar

[12]

D. KwietniakM. Lacka and P. Oprocha., A panorama of specification-like properties and their consequences, Contemporary Mathematics, 669 (2016), 155-186.  doi: 10.1090/conm/669/13428.  Google Scholar

[13]

M. Misiurewicz and J. Smítal, Smooth chaotic maps with zero topological entropy, Ergodic Theory and Dynamical Systems, 8 (1988), 421-424.  doi: 10.1017/S0143385700004557.  Google Scholar

[14]

W. Sun and X. Tian, Diffeomorphisms with various $C^1$-stable properties, Acta Mathematica Scientia, 32 (2012), 552-558.  doi: 10.1016/S0252-9602(12)60037-X.  Google Scholar

[15]

X. TianS. Wang and X. Wang, Intermediate Lyapunov exponents for system with periodic gluing orbit property, Discrete and Continuous Dynamical Systems - A, 39 (2019), 1019-1032.  doi: 10.3934/dcds.2019042.  Google Scholar

[16]

S. Xiang and Y. Zheng, Multifractal analysis for maps with the gluing orbit property, Taiwanese Journal of Mathematics, 21 (2017), 1099-1113.  doi: 10.11650/tjm/7946.  Google Scholar

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