American Institute of Mathematical Sciences

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

Minimality, distality and equicontinuity for semigroup actions on compact Hausdorff spaces

Joseph Auslander 1, and  Xiongping Dai 2,

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} $.

Keywords: Semigroup action, equicontinuity, distality, minimality, almost periodicity, reflection principle.
Mathematics Subject Classification: 37B05, 37B20, 20M20.
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. Google Scholar

[2] E. Akin, Recurrence in Topological Dynamics, Plenum Press, New York London, 1997. doi: 10.1007/978-1-4757-2668-8. Google Scholar
[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. Google Scholar

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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. Google Scholar

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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. Google Scholar

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E. Akin and S. Kolyada, Li-Yorke sensitivity, Nonlinearity, 16 (2003), 1421-1433. doi: 10.1088/0951-7715/16/4/313. Google Scholar

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J. Auslander, Minimal Flows and Their Extensions, North-Holland Math. Studies Vol. 153. North-Holland, Amsterdam, 1988. Google Scholar

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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. Google Scholar

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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. Google Scholar

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J. Auslander and N. Markley, Locally almost periodic minimal flows, J. Difference Equ. Appl., 15 (2009), 97-109. doi: 10.1080/10236190802385330. Google Scholar

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J. Auslander and J. Yorke, Interval maps, factors of maps and chaos, Tôhoku Math. J., 32 (1980), 177–188. doi: 10.2748/tmj/1178229634. Google Scholar

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F. Blanchard, B. Host and A. Maass, Topological complexity, Ergodic. Th. & Dynam. Sys., 20 (2000), 641-662. doi: 10.1017/S0143385700000341. Google Scholar

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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. doi: 10.1007/s11425-016-9139-1. Google Scholar

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M. Day, Amenable semigroups, Illinois J. Math., 1 (1957), 509-544. doi: 10.1215/ijm/1255380675. Google Scholar

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J. Dixmier, Les moyennes invariantes dans les semi-groupes et leures applications, Acta Sci. Math. Szeged., 12 (1950), 213-227. Google Scholar

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N. Dunford and J. T. Schwartz, Linear Operators, Part Ⅰ: General Theory, Interscience Publishers, Inc., New York, 1958. Google Scholar

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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. Google Scholar

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I. Namioka, Separate continuity and joint continuity, Pacific J. Math., 51 (1974), 515-531. doi: 10.2140/pjm.1974.51.515. Google Scholar

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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. Google Scholar

[2] E. Akin, Recurrence in Topological Dynamics, Plenum Press, New York London, 1997. doi: 10.1007/978-1-4757-2668-8. Google Scholar
[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[6]

E. Akin and S. Kolyada, Li-Yorke sensitivity, Nonlinearity, 16 (2003), 1421-1433. doi: 10.1088/0951-7715/16/4/313. Google Scholar

[7]

J. Auslander, Minimal Flows and Their Extensions, North-Holland Math. Studies Vol. 153. North-Holland, Amsterdam, 1988. Google Scholar

[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. Google Scholar

[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. Google Scholar

[10]

J. Auslander and N. Markley, Locally almost periodic minimal flows, J. Difference Equ. Appl., 15 (2009), 97-109. doi: 10.1080/10236190802385330. Google Scholar

[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. Google Scholar

[12]

F. Blanchard, B. Host and A. Maass, Topological complexity, Ergodic. Th. & Dynam. Sys., 20 (2000), 641-662. doi: 10.1017/S0143385700000341. Google Scholar

[13]

N. Bourbaki, Topologie Générale, Chapitre IX, Actualités Sci. Ind. No. 1045, Hermann, Paris, 1948. Google Scholar

[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. Google Scholar

[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. Google Scholar

[16]

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

[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. Google Scholar

[18]

J. P. Clay, Variations on equicontinuity, Duke Math. J., 30 (1963), 423-431. doi: 10.1215/S0012-7094-63-03045-X. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[24]

M. Day, Amenable semigroups, Illinois J. Math., 1 (1957), 509-544. doi: 10.1215/ijm/1255380675. Google Scholar

[25]

J. Dixmier, Les moyennes invariantes dans les semi-groupes et leures applications, Acta Sci. Math. Szeged., 12 (1950), 213-227. Google Scholar

[26]

N. Dunford and J. T. Schwartz, Linear Operators, Part Ⅰ: General Theory, Interscience Publishers, Inc., New York, 1958. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[31]

R. Ellis, Locally compact transformation groups, Duke Math. J., 24 (1957), 119-125. doi: 10.1215/S0012-7094-57-02417-1. Google Scholar

[32]

R. Ellis, Distal transformation groups, Pacific J. Math., 8 (1958), 401-405. doi: 10.2140/pjm.1958.8.401. Google Scholar

[33]

R. Ellis, Equicontinuity and almost periodic functions, Proc. Amer. Math. Soc., 10 (1959), 637-643. doi: 10.1090/S0002-9939-1959-0107225-X. Google Scholar

[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. Google Scholar

[35]

R. Ellis, Universal minimal sets, Proc. Amer. Math. Soc., 11 (1960), 540-543. doi: 10.1090/S0002-9939-1960-0117716-1. Google Scholar

[36]

R. Ellis, Lectures on Topological Dynamics, Benjamin, New York, 1969. Google Scholar

[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. Google Scholar

[38]

S. Ferenczi, Complexity of sequences and dynamical systems, Discrete. Math., 206 (1999), 145-154. doi: 10.1016/S0012-365X(98)00400-2. Google Scholar

[39]

M.K. Jr Fort, Category theorems, Fund. Math., 42 (1955), 276-288. doi: 10.4064/fm-42-2-276-288. Google Scholar

[40]

H. Furstenberg, The structure of distal flows, Amer. J. Math., 85 (1963), 477-515. doi: 10.2307/2373137. Google Scholar

[41] H. Furstenberg, Recurrence in Ergodic Theory and Combinatorial Number Theory, Princeton University Press, Princeton, New Jersey, 1981. Google Scholar
[42]

E. Glasner, Proximal Flows, Lecture Notes in Math., 517, Springer-Verlag, 1976. Google Scholar

[43]

E. Glasner, Ergodic Theory via Joinings, Math. Surveys and Monographs, 101, Amerian Math. Soc., 2003. doi: 10.1090/surv/101. Google Scholar

[44]

S. Glasner and D. Maon, Rigidity in topological dynamics, Ergod. Th. & Dyn. Syst., 9 (1989), 309-320. doi: 10.1017/S0143385700004983. Google Scholar

[45]

W. H. Gottschalk, Almost periodic points with respect to transformation semi-groups, Annals of Math., 47 (1946), 762-766. doi: 10.2307/1969233. Google Scholar

[46]

W. H. Gottschalk, Transitivity and equicontinuity, Bull. Amer. Math. Soc., 54 (1948), 982-984. doi: 10.1090/S0002-9904-1948-09119-4. Google Scholar

[47]

W. H. Gottschalk and G. A. Hedlund, Topological Dynamics, Amer. Math. Soc. Coll. Publ. Vol. 36, Amer. Math. Soc., Providence, R.I., 1955. Google Scholar

[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. Google Scholar

[49]

D. Helmer, Continuity of semigroup actions, Semigroup Forum, 23 (1981), 153-188. doi: 10.1007/BF02676642. Google Scholar

[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. Google Scholar

[51]

P. Kenderov, Dense strong continuity of pointwise mappings, Pacific J. Math., 89 (1980), 111-130. doi: 10.2140/pjm.1980.89.111. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[55]

D. Montgomery, Continuity in topological groups, Bull. Amer. Math. Soc., 42 (1936), 879-882. doi: 10.1090/S0002-9904-1936-06456-6. Google Scholar

[56]

I. Namioka, Separate continuity and joint continuity, Pacific J. Math., 51 (1974), 515-531. doi: 10.2140/pjm.1974.51.515. Google Scholar

[57] V. V. Nemytskii and V. V. Stepanov, Qualitative Theory of Differential Equations, Princeton University Press, Princeton, New Jersey, 1960. Google Scholar
[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. Google Scholar

[59]

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

[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. Google Scholar

[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. Google Scholar

[62]

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