doi: 10.3934/dcds.2021165
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Expansive flows on uniform spaces

Department of Mathematics, Chungnam National University, Daejeon 34134, Republic of Korea

* Corresponding author: Se-Hyun Ku

Received  March 2021 Revised  September 2021 Early access November 2021

Fund Project: The author was supported by Basic Science Research Program through the National Research Foundation of Korea(NRF) funded by the Ministry of Education(NRF-2018R1D1A1B07051286) and by the National Research Foundation of Korea(NRF) grant funded by the Korea government (MSIT) (No. 2021R1C1C2011737)

In this paper we study several dynamical properties on uniform spaces. We define expansive flows on uniform spaces and provide some equivalent ways of defining expansivity. We also define the concept of expansive measures for flows on uniform spaces. We prove for flows on compact uniform spaces that every expansive measure vanishes along the orbits and has no singularities in the support. We also prove that every expansive measure for flows on uniform spaces is aperiodic and is expansive with respect to time-$ T $ map. Furthermore we show that every expansive measure for flows on compact uniform spaces maintains expansive under topological equivalence.

Citation: Se-Hyun Ku. Expansive flows on uniform spaces. Discrete & Continuous Dynamical Systems, doi: 10.3934/dcds.2021165
References:
[1]

A. Arbieto and C. A. Morales, Some properties of positive entropy maps, Ergodic Theory Dynam. Systems, 34 (2014), 765-776.  doi: 10.1017/etds.2012.162.  Google Scholar

[2]

A. Artigue, Positive expansive flows, Topology Appl., 165 (2014), 121-132.  doi: 10.1016/j.topol.2014.01.015.  Google Scholar

[3]

V. I. Bogachev, Measure Theory, Vol. I, II, Springer-Verlag, Berlin, 2007. doi: 10.1007/978-3-540-34514-5.  Google Scholar

[4]

R. Bowen and P. Walters, Expansive one-parameter flows, J. Differential Equations, 12 (1972), 180-193.  doi: 10.1016/0022-0396(72)90013-7.  Google Scholar

[5]

B. F. Bryant, On expansive homeomorphisms, Pacific J. Math., 10 (1960), 1163-1167.  doi: 10.2140/pjm.1960.10.1163.  Google Scholar

[6]

D. Carrasco-Olivera and C. A. Morales, Expansive measures for flows, J. Differential Equations, 256 (2014), 2246-2260.  doi: 10.1016/j.jde.2013.12.019.  Google Scholar

[7]

M. Cerminara and M. Sambarino, Stable and unstable sets of $C^0$ perturbations of expansive homeomorphisms of surfaces, Nonlinearity, 12 (1999), 321-332.  doi: 10.1088/0951-7715/12/2/011.  Google Scholar

[8]

M. Eisenberg, Expansive transformation semigroups of endomorphisms, Fund. Math., 59 (1966), 313-321.  doi: 10.4064/fm-59-3-313-321.  Google Scholar

[9]

W. H. Gottschalk and G. A. Hedlund, Topological Dynamics, American Mathematical Society Colloquium Publications, Vol. 36, American Mathematical Society, Providence, R. I., 1955.  Google Scholar

[10]

A. A. Gura, The horocycle flow on a surface of negative curvature is separating, Mat. Zametki, 36 (1984), 279-284.   Google Scholar

[11]

J. F. Jakobsen and W. R. Utz, The non-existence of expansive homeomorphisms on a closed $2$-cell, Pacific J. Math., 10 (1960), 1319-1321.  doi: 10.2140/pjm.1960.10.1319.  Google Scholar

[12]

I. M. James, Introduction to Uniform Spaces, London Mathematical Society Lecture Note Series, 144, Cambridge University Press, Cambridge, 1990. doi: 10.1017/CBO9780511721519.  Google Scholar

[13]

H. Kato, Expansive homeomorphisms on surfaces with holes, Topology Appl., 82 (1998), 267-277.  doi: 10.1016/S0166-8641(97)00069-2.  Google Scholar

[14]

J. L. Kelley, General Topology, Graduate Texts in Mathematics, No. 27, Springer-Verlag, New York-Berlin, 1975.  Google Scholar

[15]

R. Mañé, Expansive homeomorphisms and topological dimension, Trans. Amer. Math. Soc., 252 (1979), 313-319.  doi: 10.1090/S0002-9947-1979-0534124-9.  Google Scholar

[16]

C. A. Morales and V. Sirvent, Expansivity for measures on uniform spaces, Trans. Amer. Math. Soc., 368 (2016), 5399-5414.  doi: 10.1090/tran/6555.  Google Scholar

[17]

C. A. Morales and V. F. Sirvent, Expansive Measures, IMPA Mathematical Publications, 29th Brazilian Mathematics Colloquium, Rio de Janeiro, 2013.  Google Scholar

[18]

T. O'Brien, Expansive homeomorphisms on compact manifolds, Proc. Amer. Math. Soc., 24 (1970), 767-771.  doi: 10.1090/S0002-9939-1970-0253308-8.  Google Scholar

[19]

R. O. Ruggiero, Expansive dynamics and hyperbolic geometry, Bol. Soc. Brasil. Mat. (N.S.), 25 (1994), 139-172.  doi: 10.1007/BF01321305.  Google Scholar

[20]

K. Sakai, Hyperbolic metrics of expansive homeomorphisms, Topology Appl., 63 (1995), 263-266.  doi: 10.1016/0166-8641(95)00083-S.  Google Scholar

[21]

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

[22]

A. Weil, Sur Les Espaces à Structure Uniforme et sur la Topologie Générale, Hermann, Paris, 1937.  Google Scholar

show all references

References:
[1]

A. Arbieto and C. A. Morales, Some properties of positive entropy maps, Ergodic Theory Dynam. Systems, 34 (2014), 765-776.  doi: 10.1017/etds.2012.162.  Google Scholar

[2]

A. Artigue, Positive expansive flows, Topology Appl., 165 (2014), 121-132.  doi: 10.1016/j.topol.2014.01.015.  Google Scholar

[3]

V. I. Bogachev, Measure Theory, Vol. I, II, Springer-Verlag, Berlin, 2007. doi: 10.1007/978-3-540-34514-5.  Google Scholar

[4]

R. Bowen and P. Walters, Expansive one-parameter flows, J. Differential Equations, 12 (1972), 180-193.  doi: 10.1016/0022-0396(72)90013-7.  Google Scholar

[5]

B. F. Bryant, On expansive homeomorphisms, Pacific J. Math., 10 (1960), 1163-1167.  doi: 10.2140/pjm.1960.10.1163.  Google Scholar

[6]

D. Carrasco-Olivera and C. A. Morales, Expansive measures for flows, J. Differential Equations, 256 (2014), 2246-2260.  doi: 10.1016/j.jde.2013.12.019.  Google Scholar

[7]

M. Cerminara and M. Sambarino, Stable and unstable sets of $C^0$ perturbations of expansive homeomorphisms of surfaces, Nonlinearity, 12 (1999), 321-332.  doi: 10.1088/0951-7715/12/2/011.  Google Scholar

[8]

M. Eisenberg, Expansive transformation semigroups of endomorphisms, Fund. Math., 59 (1966), 313-321.  doi: 10.4064/fm-59-3-313-321.  Google Scholar

[9]

W. H. Gottschalk and G. A. Hedlund, Topological Dynamics, American Mathematical Society Colloquium Publications, Vol. 36, American Mathematical Society, Providence, R. I., 1955.  Google Scholar

[10]

A. A. Gura, The horocycle flow on a surface of negative curvature is separating, Mat. Zametki, 36 (1984), 279-284.   Google Scholar

[11]

J. F. Jakobsen and W. R. Utz, The non-existence of expansive homeomorphisms on a closed $2$-cell, Pacific J. Math., 10 (1960), 1319-1321.  doi: 10.2140/pjm.1960.10.1319.  Google Scholar

[12]

I. M. James, Introduction to Uniform Spaces, London Mathematical Society Lecture Note Series, 144, Cambridge University Press, Cambridge, 1990. doi: 10.1017/CBO9780511721519.  Google Scholar

[13]

H. Kato, Expansive homeomorphisms on surfaces with holes, Topology Appl., 82 (1998), 267-277.  doi: 10.1016/S0166-8641(97)00069-2.  Google Scholar

[14]

J. L. Kelley, General Topology, Graduate Texts in Mathematics, No. 27, Springer-Verlag, New York-Berlin, 1975.  Google Scholar

[15]

R. Mañé, Expansive homeomorphisms and topological dimension, Trans. Amer. Math. Soc., 252 (1979), 313-319.  doi: 10.1090/S0002-9947-1979-0534124-9.  Google Scholar

[16]

C. A. Morales and V. Sirvent, Expansivity for measures on uniform spaces, Trans. Amer. Math. Soc., 368 (2016), 5399-5414.  doi: 10.1090/tran/6555.  Google Scholar

[17]

C. A. Morales and V. F. Sirvent, Expansive Measures, IMPA Mathematical Publications, 29th Brazilian Mathematics Colloquium, Rio de Janeiro, 2013.  Google Scholar

[18]

T. O'Brien, Expansive homeomorphisms on compact manifolds, Proc. Amer. Math. Soc., 24 (1970), 767-771.  doi: 10.1090/S0002-9939-1970-0253308-8.  Google Scholar

[19]

R. O. Ruggiero, Expansive dynamics and hyperbolic geometry, Bol. Soc. Brasil. Mat. (N.S.), 25 (1994), 139-172.  doi: 10.1007/BF01321305.  Google Scholar

[20]

K. Sakai, Hyperbolic metrics of expansive homeomorphisms, Topology Appl., 63 (1995), 263-266.  doi: 10.1016/0166-8641(95)00083-S.  Google Scholar

[21]

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

[22]

A. Weil, Sur Les Espaces à Structure Uniforme et sur la Topologie Générale, Hermann, Paris, 1937.  Google Scholar

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