March  2022, 42(3): 1585-1598. doi: 10.3934/dcds.2021165

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 Published  March 2022 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 and Continuous Dynamical Systems, 2022, 42 (3) : 1585-1598. 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.

[2]

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

[3]

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

[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.

[5]

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

[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.

[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.

[8]

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

[9]

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

[10]

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

[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.

[12]

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

[13]

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

[14]

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

[15]

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

[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.

[17]

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

[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.

[19]

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

[20]

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

[21]

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

[22]

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

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.

[2]

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

[3]

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

[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.

[5]

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

[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.

[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.

[8]

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

[9]

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

[10]

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

[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.

[12]

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

[13]

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

[14]

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

[15]

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

[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.

[17]

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

[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.

[19]

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

[20]

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

[21]

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

[22]

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

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