August  2019, 39(8): 4647-4711. doi: 10.3934/dcds.2019190

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

To the memory of Professor Isaac Namioka

Received  September 2018 Revised  March 2019 Published  May 2019

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} $.

Citation: Joseph Auslander, Xiongping Dai. Minimality, distality and equicontinuity for semigroup actions on compact Hausdorff spaces. Discrete & Continuous Dynamical Systems - A, 2019, 39 (8) : 4647-4711. doi: 10.3934/dcds.2019190
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. AkinJ. 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. BlanchardB. 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.

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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.

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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.

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H. Chu, On universal minimal transformation groups, Illinois J. Math., 6 (1962), 317-326. doi: 10.1215/ijm/1255632329.

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J. P. Clay, Proximity relations in transformation groups, Trans. Amer. Math. Soc., 108 (1963), 88-96. doi: 10.1090/S0002-9947-1963-0154269-3.

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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.

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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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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.

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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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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.

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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.

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D. EllisR. 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.

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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.

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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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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.

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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]

H. L. Royden, Real Analysis, 3rd ed. MacMillan, NY, 1988.

[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. doi: 10.1090/memo/0190.

[62]

G. R. SellW. Shen and Y. Yi, Topological dynamics and differential equations, Contemp. Math., 215 (1998), 279-297. doi: 10.1090/conm/215/02948.

[63]

W. Shen and Y. Yi, Almost automorphic and almost periodic dynamics in skew-product semiflows, Mem. Amer. Math. Soc., 136 (1998), x+93 pp. doi: 10.1090/memo/0647.

[64]

J.-P. Troallic, Espaces fonctionnels et théorèmes de I. Namioka, Bull. Soc. Math. France, 107 (1979), 127-137.

[65]

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. doi: 10.1007/BFb0062032.

[66]

S. L. Troyanski, On locally uniformly convex and differentiable norms in certain nonseparable Banach spaces, Studia Math., 37 (1971), 173-180. doi: 10.4064/sm-37-2-173-180.

[67]

W. A. Veech, Point-distal flows, Amer. J. Math., 92 (1970), 205-242. doi: 10.2307/2373504.

[68]

W. A. Veech, Topological dynamics, Bull. Amer. Math. Soc., 83 (1977), 775-830. doi: 10.1090/S0002-9904-1977-14319-X.

[69]

Z. Wang and G. Zhang, Chaotic behavior of group actions, Contemp. Math., 669 (2016), 299-315. doi: 10.1090/conm/669/13434.

show all references

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. AkinJ. 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. BlanchardB. 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. EllisR. 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]

H. L. Royden, Real Analysis, 3rd ed. MacMillan, NY, 1988.

[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. doi: 10.1090/memo/0190.

[62]

G. R. SellW. Shen and Y. Yi, Topological dynamics and differential equations, Contemp. Math., 215 (1998), 279-297. doi: 10.1090/conm/215/02948.

[63]

W. Shen and Y. Yi, Almost automorphic and almost periodic dynamics in skew-product semiflows, Mem. Amer. Math. Soc., 136 (1998), x+93 pp. doi: 10.1090/memo/0647.

[64]

J.-P. Troallic, Espaces fonctionnels et théorèmes de I. Namioka, Bull. Soc. Math. France, 107 (1979), 127-137.

[65]

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. doi: 10.1007/BFb0062032.

[66]

S. L. Troyanski, On locally uniformly convex and differentiable norms in certain nonseparable Banach spaces, Studia Math., 37 (1971), 173-180. doi: 10.4064/sm-37-2-173-180.

[67]

W. A. Veech, Point-distal flows, Amer. J. Math., 92 (1970), 205-242. doi: 10.2307/2373504.

[68]

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