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Chern-Simons gauged sigma model into $ \mathbb{H}^2 $ and its self-dual equations
Minimality, distality and equicontinuity for semigroup actions on compact Hausdorff spaces
1. | Department of Mathematics, University of Maryland, College Park, MD 20742, USA |
2. | Department of Mathematics, Nanjing University, Nanjing 210093, China |
Let $ T $ be any topological semigroup and $ (T, X) $ with phase mapping $ (t, x)\mapsto tx $ a semiflow on a compact $ \text{T}_2 $ space $ X $. If $ tX = X $ for all $ t $ in $ T $ then $ (T, X) $ is called surjective; if $ x\mapsto tx $, for each $ t $ in $ T $, is 1-1 onto, then $ (T, X) $ is termed invertible and the latter induces a right-action semiflow $ (X, T) $ with the phase mapping $ (x, t)\mapsto xt: = t^{-1}x $. We show that $ (T, X) $ is equicontinuous surjective iff it is uniformly distal iff $ (X, T) $ is equicontinuous surjective. We then consider minimality, distality, point-distality, and sensitivity of $ (X, T) $ when $ (T, X) $ possesses these dynamics. We also study the pointwise recurrence and Gottschalk's weak almost periodicity of flow on a zero-dimensional space with phase group $ \mathbb{Z} $.
References:
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E. Akin,
On chain continuity, Discret. Contin. Dyn. Syst., 2 (1996), 111-120.
doi: 10.3934/dcds.1996.2.111. |
[2] |
E. Akin, Recurrence in Topological Dynamics, Plenum Press, New York London, 1997.
doi: 10.1007/978-1-4757-2668-8.![]() ![]() ![]() |
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E. Akin and J. Auslander,
Almost periodic sets and subactions in topological dynamics, Proc. Amer. Math. Soc., 131 (2003), 3059-3062.
doi: 10.1090/S0002-9939-03-07005-9. |
[4] |
E. Akin, J. Auslander and K. Berg, When is a transitive map chaotic?, in: Conference in Ergodic Theory and Probability, Columbus, OH, 1993, in: Ohio State Univ. Math. Res. Inst. Publ., 5, de Gruyter, Berlin, 1996, 25–40. |
[5] |
E. Akin, J. Auslander and K. Berg,
Almost equicontinuity and the enveloping semigroup, Contemporary Math., 215 (1998), 75-81.
doi: 10.1090/conm/215/02931. |
[6] |
E. Akin and S. Kolyada,
Li-Yorke sensitivity, Nonlinearity, 16 (2003), 1421-1433.
doi: 10.1088/0951-7715/16/4/313. |
[7] |
J. Auslander, Minimal Flows and Their Extensions, North-Holland Math. Studies Vol. 153. North-Holland, Amsterdam, 1988. |
[8] |
J. Auslander and H. Furstenberg,
Product recurrence and distal points, Trans. Amer. Math. Soc., 343 (1994), 221-232.
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[9] |
J. Auslander and F. Hahn,
Point transitive flows, algebras of functions, and the Bebutov system, Fund. Math., 60 (1967), 117-137.
doi: 10.4064/fm-60-2-117-137. |
[10] |
J. Auslander and N. Markley,
Locally almost periodic minimal flows, J. Difference Equ. Appl., 15 (2009), 97-109.
doi: 10.1080/10236190802385330. |
[11] |
J. Auslander and J. Yorke, Interval maps, factors of maps and chaos, Tôhoku Math. J., 32 (1980), 177–188.
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F. Blanchard, B. Host and A. Maass,
Topological complexity, Ergodic. Th. & Dynam. Sys., 20 (2000), 641-662.
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On uniformly recurrent motions of topological semigroup actions, Discret. Contin. Dyn. Syst., 36 (2016), 2931-2944.
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Joint continuity of separately continuous functions, Proc. Amer. Math. Soc., 82 (1981), 455-461.
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H. Chu,
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Proximity relations in transformation groups, Trans. Amer. Math. Soc., 108 (1963), 88-96.
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Variations on equicontinuity, Duke Math. J., 30 (1963), 423-431.
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X. Dai and H. Liang,
On Galvin's theorem for compact Hausdorff right-topological semigroups with dense topological centers, Sci. China Math., 60 (2017), 2421-2428.
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Devaney chaos, Li-Yorke chaos, and multi-dimensional Li-Yorke chaos for topological dynamics, J. Differential Equations, 263 (2017), 5521-5553.
doi: 10.1016/j.jde.2017.06.021. |
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X. Dai and Z. Xiao,
Equicontinuity, uniform almost periodicity, and regionally proximal relation for topological semiflows, Topology Appl., 231 (2017), 35-49.
doi: 10.1016/j.topol.2017.09.007. |
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Means for bounded functions and ergodicity of the bounded representations of semigroups, Trans. Amer. Math. Soc., 69 (1950), 276-291.
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M. Day,
Amenable semigroups, Illinois J. Math., 1 (1957), 509-544.
doi: 10.1215/ijm/1255380675. |
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J. Dixmier,
Les moyennes invariantes dans les semi-groupes et leures applications, Acta Sci. Math. Szeged., 12 (1950), 213-227.
|
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N. Dunford and J. T. Schwartz, Linear Operators, Part Ⅰ: General Theory, Interscience Publishers, Inc., New York, 1958. |
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Abstract ergodic theorems and weak almost periodic functions, Trans. Amer. Math. Soc., 67 (1949), 217-240.
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D. Ellis, R. Ellis and M. Nerurkar,
The topological dynamics of semigroup actions, Trans. Amer. Math. Soc., 353 (2001), 1279-1320.
doi: 10.1090/S0002-9947-00-02704-5. |
[30] |
R. Ellis,
A note on the continuity of the inverse, Proc. Amer. Math. Soc., 8 (1957), 372-373.
doi: 10.1090/S0002-9939-1957-0083681-9. |
[31] |
R. Ellis,
Locally compact transformation groups, Duke Math. J., 24 (1957), 119-125.
doi: 10.1215/S0012-7094-57-02417-1. |
[32] |
R. Ellis,
Distal transformation groups, Pacific J. Math., 8 (1958), 401-405.
doi: 10.2140/pjm.1958.8.401. |
[33] |
R. Ellis,
Equicontinuity and almost periodic functions, Proc. Amer. Math. Soc., 10 (1959), 637-643.
doi: 10.1090/S0002-9939-1959-0107225-X. |
[34] |
R. Ellis,
A semigroup associated with a transformation group, Trans. Amer. Math. Soc., 94 (1960), 272-281.
doi: 10.1090/S0002-9947-1960-0123636-3. |
[35] |
R. Ellis,
Universal minimal sets, Proc. Amer. Math. Soc., 11 (1960), 540-543.
doi: 10.1090/S0002-9939-1960-0117716-1. |
[36] |
R. Ellis, Lectures on Topological Dynamics, Benjamin, New York, 1969. |
[37] |
R. Ellis and M. Nerurkar,
Weakly almost periodic flows, Trans. Amer. Math. Soc., 313 (1989), 103-119.
doi: 10.1090/S0002-9947-1989-0930084-3. |
[38] |
S. Ferenczi,
Complexity of sequences and dynamical systems, Discrete. Math., 206 (1999), 145-154.
doi: 10.1016/S0012-365X(98)00400-2. |
[39] |
M.K. Jr Fort,
Category theorems, Fund. Math., 42 (1955), 276-288.
doi: 10.4064/fm-42-2-276-288. |
[40] |
H. Furstenberg,
The structure of distal flows, Amer. J. Math., 85 (1963), 477-515.
doi: 10.2307/2373137. |
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H. Furstenberg, Recurrence in Ergodic Theory and Combinatorial Number Theory, Princeton University Press, Princeton, New Jersey, 1981.
![]() ![]() |
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E. Glasner, Proximal Flows, Lecture Notes in Math., 517, Springer-Verlag, 1976. |
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E. Glasner, Ergodic Theory via Joinings, Math. Surveys and Monographs, 101, Amerian Math. Soc., 2003.
doi: 10.1090/surv/101. |
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S. Glasner and D. Maon,
Rigidity in topological dynamics, Ergod. Th. & Dyn. Syst., 9 (1989), 309-320.
doi: 10.1017/S0143385700004983. |
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W. H. Gottschalk,
Almost periodic points with respect to transformation semi-groups, Annals of Math., 47 (1946), 762-766.
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W. H. Gottschalk,
Transitivity and equicontinuity, Bull. Amer. Math. Soc., 54 (1948), 982-984.
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W. H. Gottschalk and G. A. Hedlund, Topological Dynamics, Amer. Math. Soc. Coll. Publ. Vol. 36, Amer. Math. Soc., Providence, R.I., 1955. |
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Points de continuite a gauche d'une action de semigroupe, Semigroup Forum, 26 (1983), 205-214.
doi: 10.1007/BF02572831. |
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D. Helmer,
Continuity of semigroup actions, Semigroup Forum, 23 (1981), 153-188.
doi: 10.1007/BF02676642. |
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J. L. Kelley, General Topology, Reprint of the 1955 edition [Van Nostrand, Toronto, Ont.]. Graduate Texts in Mathematics, No. 27. Springer-Verlag, New York-Berlin, 1975. |
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P. Kenderov,
Dense strong continuity of pointwise mappings, Pacific J. Math., 89 (1980), 111-130.
doi: 10.2140/pjm.1980.89.111. |
[52] |
E. Kontorovich and M. Megrelishvili,
A note on sensitivity of semigroup actions, Semigroup Forum, 76 (2008), 133-141.
doi: 10.1007/s00233-007-9033-5. |
[53] |
J. D. Lawson,
Points of continuity for semigroup actions, Trans. Amer. Math. Soc., 284 (1984), 183-202.
doi: 10.1090/S0002-9947-1984-0742420-7. |
[54] |
D. McMahon and L. Nachman,
An intrinsic characterization for PI flows, Pacific J. Math., 89 (1980), 391-403.
doi: 10.2140/pjm.1980.89.391. |
[55] |
D. Montgomery,
Continuity in topological groups, Bull. Amer. Math. Soc., 42 (1936), 879-882.
doi: 10.1090/S0002-9904-1936-06456-6. |
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I. Namioka,
Separate continuity and joint continuity, Pacific J. Math., 51 (1974), 515-531.
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V. V. Nemytskii and V. V. Stepanov, Qualitative Theory of Differential Equations, Princeton University Press, Princeton, New Jersey, 1960.
![]() ![]() |
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P. Oprocha and G. Zhang,
On local aspects of topological weak mixing in dimension one and beyond, Studia. Math., 202 (2011), 261-288.
doi: 10.4064/sm202-3-4. |
[59] | |
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R. J. Sacker and G. R. Sell,
Finite extension of minimal transformations groups, Trans. Amer. Math. Soc., 190 (1974), 325-334.
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R. J. Sacker and G. R. Sell, Lifting properties in skew-product flows with applications to differential equations, Mem. Amer. Math. Soc., 11 (1977), ⅳ+67 pp.
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J.-P. Troallic,
Espaces fonctionnels et théorèmes de I. Namioka, Bull. Soc. Math. France, 107 (1979), 127-137.
|
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J.-P. Troallic, Semigroupes semitopologiques et presqueperiodicite, (Proc. Conf. Semigroups, Oberwolfach 1981), Lecture Notes in Math., 998, Springer-Verlag, Berlin New York, 1983,239–251.
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W. A. Veech,
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show all references
To the memory of Professor Isaac Namioka
References:
[1] |
E. Akin,
On chain continuity, Discret. Contin. Dyn. Syst., 2 (1996), 111-120.
doi: 10.3934/dcds.1996.2.111. |
[2] |
E. Akin, Recurrence in Topological Dynamics, Plenum Press, New York London, 1997.
doi: 10.1007/978-1-4757-2668-8.![]() ![]() ![]() |
[3] |
E. Akin and J. Auslander,
Almost periodic sets and subactions in topological dynamics, Proc. Amer. Math. Soc., 131 (2003), 3059-3062.
doi: 10.1090/S0002-9939-03-07005-9. |
[4] |
E. Akin, J. Auslander and K. Berg, When is a transitive map chaotic?, in: Conference in Ergodic Theory and Probability, Columbus, OH, 1993, in: Ohio State Univ. Math. Res. Inst. Publ., 5, de Gruyter, Berlin, 1996, 25–40. |
[5] |
E. Akin, J. Auslander and K. Berg,
Almost equicontinuity and the enveloping semigroup, Contemporary Math., 215 (1998), 75-81.
doi: 10.1090/conm/215/02931. |
[6] |
E. Akin and S. Kolyada,
Li-Yorke sensitivity, Nonlinearity, 16 (2003), 1421-1433.
doi: 10.1088/0951-7715/16/4/313. |
[7] |
J. Auslander, Minimal Flows and Their Extensions, North-Holland Math. Studies Vol. 153. North-Holland, Amsterdam, 1988. |
[8] |
J. Auslander and H. Furstenberg,
Product recurrence and distal points, Trans. Amer. Math. Soc., 343 (1994), 221-232.
doi: 10.1090/S0002-9947-1994-1170562-X. |
[9] |
J. Auslander and F. Hahn,
Point transitive flows, algebras of functions, and the Bebutov system, Fund. Math., 60 (1967), 117-137.
doi: 10.4064/fm-60-2-117-137. |
[10] |
J. Auslander and N. Markley,
Locally almost periodic minimal flows, J. Difference Equ. Appl., 15 (2009), 97-109.
doi: 10.1080/10236190802385330. |
[11] |
J. Auslander and J. Yorke, Interval maps, factors of maps and chaos, Tôhoku Math. J., 32 (1980), 177–188.
doi: 10.2748/tmj/1178229634. |
[12] |
F. Blanchard, B. Host and A. Maass,
Topological complexity, Ergodic. Th. & Dynam. Sys., 20 (2000), 641-662.
doi: 10.1017/S0143385700000341. |
[13] |
N. Bourbaki, Topologie Générale, Chapitre IX, Actualités Sci. Ind. No. 1045, Hermann, Paris, 1948. |
[14] |
B. Chen and X. Dai,
On uniformly recurrent motions of topological semigroup actions, Discret. Contin. Dyn. Syst., 36 (2016), 2931-2944.
doi: 10.3934/dcds.2016.36.2931. |
[15] |
J. P. R. Christensen,
Joint continuity of separately continuous functions, Proc. Amer. Math. Soc., 82 (1981), 455-461.
doi: 10.1090/S0002-9939-1981-0612739-1. |
[16] |
H. Chu,
On universal minimal transformation groups, Illinois J. Math., 6 (1962), 317-326.
doi: 10.1215/ijm/1255632329. |
[17] |
J. P. Clay,
Proximity relations in transformation groups, Trans. Amer. Math. Soc., 108 (1963), 88-96.
doi: 10.1090/S0002-9947-1963-0154269-3. |
[18] |
J. P. Clay,
Variations on equicontinuity, Duke Math. J., 30 (1963), 423-431.
doi: 10.1215/S0012-7094-63-03045-X. |
[19] |
X. Dai and E. Glasner,
On universal minimal proximal flows of topological groups, Proc. Amer. Math. Soc., 147 (2019), 1149-1164.
doi: 10.1090/proc/14292. |
[20] |
X. Dai and H. Liang,
On Galvin's theorem for compact Hausdorff right-topological semigroups with dense topological centers, Sci. China Math., 60 (2017), 2421-2428.
doi: 10.1007/s11425-016-9139-1. |
[21] |
X. Dai and X. Tang,
Devaney chaos, Li-Yorke chaos, and multi-dimensional Li-Yorke chaos for topological dynamics, J. Differential Equations, 263 (2017), 5521-5553.
doi: 10.1016/j.jde.2017.06.021. |
[22] |
X. Dai and Z. Xiao,
Equicontinuity, uniform almost periodicity, and regionally proximal relation for topological semiflows, Topology Appl., 231 (2017), 35-49.
doi: 10.1016/j.topol.2017.09.007. |
[23] |
M. Day,
Means for bounded functions and ergodicity of the bounded representations of semigroups, Trans. Amer. Math. Soc., 69 (1950), 276-291.
doi: 10.1090/S0002-9947-1950-0044031-5. |
[24] |
M. Day,
Amenable semigroups, Illinois J. Math., 1 (1957), 509-544.
doi: 10.1215/ijm/1255380675. |
[25] |
J. Dixmier,
Les moyennes invariantes dans les semi-groupes et leures applications, Acta Sci. Math. Szeged., 12 (1950), 213-227.
|
[26] |
N. Dunford and J. T. Schwartz, Linear Operators, Part Ⅰ: General Theory, Interscience Publishers, Inc., New York, 1958. |
[27] |
F. Eberlein,
Abstract ergodic theorems and weak almost periodic functions, Trans. Amer. Math. Soc., 67 (1949), 217-240.
doi: 10.1090/S0002-9947-1949-0036455-9. |
[28] |
D. Ellis and R. Ellis, Automorphisms and Equivalence Relations in Topological Dynamics, London Math. Soc. Lecture Note Ser., 412, Cambridge Univ. Press, 2014.
doi: 10.1017/CBO9781107416253. |
[29] |
D. Ellis, R. Ellis and M. Nerurkar,
The topological dynamics of semigroup actions, Trans. Amer. Math. Soc., 353 (2001), 1279-1320.
doi: 10.1090/S0002-9947-00-02704-5. |
[30] |
R. Ellis,
A note on the continuity of the inverse, Proc. Amer. Math. Soc., 8 (1957), 372-373.
doi: 10.1090/S0002-9939-1957-0083681-9. |
[31] |
R. Ellis,
Locally compact transformation groups, Duke Math. J., 24 (1957), 119-125.
doi: 10.1215/S0012-7094-57-02417-1. |
[32] |
R. Ellis,
Distal transformation groups, Pacific J. Math., 8 (1958), 401-405.
doi: 10.2140/pjm.1958.8.401. |
[33] |
R. Ellis,
Equicontinuity and almost periodic functions, Proc. Amer. Math. Soc., 10 (1959), 637-643.
doi: 10.1090/S0002-9939-1959-0107225-X. |
[34] |
R. Ellis,
A semigroup associated with a transformation group, Trans. Amer. Math. Soc., 94 (1960), 272-281.
doi: 10.1090/S0002-9947-1960-0123636-3. |
[35] |
R. Ellis,
Universal minimal sets, Proc. Amer. Math. Soc., 11 (1960), 540-543.
doi: 10.1090/S0002-9939-1960-0117716-1. |
[36] |
R. Ellis, Lectures on Topological Dynamics, Benjamin, New York, 1969. |
[37] |
R. Ellis and M. Nerurkar,
Weakly almost periodic flows, Trans. Amer. Math. Soc., 313 (1989), 103-119.
doi: 10.1090/S0002-9947-1989-0930084-3. |
[38] |
S. Ferenczi,
Complexity of sequences and dynamical systems, Discrete. Math., 206 (1999), 145-154.
doi: 10.1016/S0012-365X(98)00400-2. |
[39] |
M.K. Jr Fort,
Category theorems, Fund. Math., 42 (1955), 276-288.
doi: 10.4064/fm-42-2-276-288. |
[40] |
H. Furstenberg,
The structure of distal flows, Amer. J. Math., 85 (1963), 477-515.
doi: 10.2307/2373137. |
[41] |
H. Furstenberg, Recurrence in Ergodic Theory and Combinatorial Number Theory, Princeton University Press, Princeton, New Jersey, 1981.
![]() ![]() |
[42] |
E. Glasner, Proximal Flows, Lecture Notes in Math., 517, Springer-Verlag, 1976. |
[43] |
E. Glasner, Ergodic Theory via Joinings, Math. Surveys and Monographs, 101, Amerian Math. Soc., 2003.
doi: 10.1090/surv/101. |
[44] |
S. Glasner and D. Maon,
Rigidity in topological dynamics, Ergod. Th. & Dyn. Syst., 9 (1989), 309-320.
doi: 10.1017/S0143385700004983. |
[45] |
W. H. Gottschalk,
Almost periodic points with respect to transformation semi-groups, Annals of Math., 47 (1946), 762-766.
doi: 10.2307/1969233. |
[46] |
W. H. Gottschalk,
Transitivity and equicontinuity, Bull. Amer. Math. Soc., 54 (1948), 982-984.
doi: 10.1090/S0002-9904-1948-09119-4. |
[47] |
W. H. Gottschalk and G. A. Hedlund, Topological Dynamics, Amer. Math. Soc. Coll. Publ. Vol. 36, Amer. Math. Soc., Providence, R.I., 1955. |
[48] |
G. Hansel and J.-P. Troallic,
Points de continuite a gauche d'une action de semigroupe, Semigroup Forum, 26 (1983), 205-214.
doi: 10.1007/BF02572831. |
[49] |
D. Helmer,
Continuity of semigroup actions, Semigroup Forum, 23 (1981), 153-188.
doi: 10.1007/BF02676642. |
[50] |
J. L. Kelley, General Topology, Reprint of the 1955 edition [Van Nostrand, Toronto, Ont.]. Graduate Texts in Mathematics, No. 27. Springer-Verlag, New York-Berlin, 1975. |
[51] |
P. Kenderov,
Dense strong continuity of pointwise mappings, Pacific J. Math., 89 (1980), 111-130.
doi: 10.2140/pjm.1980.89.111. |
[52] |
E. Kontorovich and M. Megrelishvili,
A note on sensitivity of semigroup actions, Semigroup Forum, 76 (2008), 133-141.
doi: 10.1007/s00233-007-9033-5. |
[53] |
J. D. Lawson,
Points of continuity for semigroup actions, Trans. Amer. Math. Soc., 284 (1984), 183-202.
doi: 10.1090/S0002-9947-1984-0742420-7. |
[54] |
D. McMahon and L. Nachman,
An intrinsic characterization for PI flows, Pacific J. Math., 89 (1980), 391-403.
doi: 10.2140/pjm.1980.89.391. |
[55] |
D. Montgomery,
Continuity in topological groups, Bull. Amer. Math. Soc., 42 (1936), 879-882.
doi: 10.1090/S0002-9904-1936-06456-6. |
[56] |
I. Namioka,
Separate continuity and joint continuity, Pacific J. Math., 51 (1974), 515-531.
doi: 10.2140/pjm.1974.51.515. |
[57] |
V. V. Nemytskii and V. V. Stepanov, Qualitative Theory of Differential Equations, Princeton University Press, Princeton, New Jersey, 1960.
![]() ![]() |
[58] |
P. Oprocha and G. Zhang,
On local aspects of topological weak mixing in dimension one and beyond, Studia. Math., 202 (2011), 261-288.
doi: 10.4064/sm202-3-4. |
[59] | |
[60] |
R. J. Sacker and G. R. Sell,
Finite extension of minimal transformations groups, Trans. Amer. Math. Soc., 190 (1974), 325-334.
doi: 10.1090/S0002-9947-1974-0350715-8. |
[61] |
R. J. Sacker and G. R. Sell, Lifting properties in skew-product flows with applications to differential equations, Mem. Amer. Math. Soc., 11 (1977), ⅳ+67 pp.
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