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

Previous Article
Chern-Simons gauged sigma model into $ \mathbb{H}^2 $ and its self-dual equations
DCDS Home
This Issue
Next Article
Steady periodic equatorial water waves with vorticity

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.
[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.
[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.
[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]	G. R. Sell, W. Shen and Y. Yi, Topological dynamics and differential equations, Contemp. Math., 215 (1998), 279-297. doi: 10.1090/conm/215/02948. Google Scholar
[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. Google Scholar
[64]	J.-P. Troallic, Espaces fonctionnels et théorèmes de I. Namioka, Bull. Soc. Math. France, 107 (1979), 127-137. Google Scholar
[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. Google Scholar
[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. Google Scholar
[67]	W. A. Veech, Point-distal flows, Amer. J. Math., 92 (1970), 205-242. doi: 10.2307/2373504. Google Scholar
[68]	W. A. Veech, Topological dynamics, Bull. Amer. Math. Soc., 83 (1977), 775-830. doi: 10.1090/S0002-9904-1977-14319-X. Google Scholar
[69]	Z. Wang and G. Zhang, Chaotic behavior of group actions, Contemp. Math., 669 (2016), 299-315. doi: 10.1090/conm/669/13434. Google Scholar

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.
[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.
[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.
[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]	G. R. Sell, W. Shen and Y. Yi, Topological dynamics and differential equations, Contemp. Math., 215 (1998), 279-297. doi: 10.1090/conm/215/02948. Google Scholar
[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. Google Scholar
[64]	J.-P. Troallic, Espaces fonctionnels et théorèmes de I. Namioka, Bull. Soc. Math. France, 107 (1979), 127-137. Google Scholar
[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. Google Scholar
[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. Google Scholar
[67]	W. A. Veech, Point-distal flows, Amer. J. Math., 92 (1970), 205-242. doi: 10.2307/2373504. Google Scholar
[68]	W. A. Veech, Topological dynamics, Bull. Amer. Math. Soc., 83 (1977), 775-830. doi: 10.1090/S0002-9904-1977-14319-X. Google Scholar
[69]	Z. Wang and G. Zhang, Chaotic behavior of group actions, Contemp. Math., 669 (2016), 299-315. doi: 10.1090/conm/669/13434. Google Scholar

[1]	Noriaki Yamazaki. Almost periodicity of solutions to free boundary problems. Conference Publications, 2001, 2001 (Special) : 386-397. doi: 10.3934/proc.2001.2001.386
[2]	Rainer Steinwandt, Adriana Suárez Corona. Cryptanalysis of a 2-party key establishment based on a semigroup action problem. Advances in Mathematics of Communications, 2011, 5 (1) : 87-92. doi: 10.3934/amc.2011.5.87
[3]	Carlos Lizama, Luz Roncal. Hölder-Lebesgue regularity and almost periodicity for semidiscrete equations with a fractional Laplacian. Discrete & Continuous Dynamical Systems - A, 2018, 38 (3) : 1365-1403. doi: 10.3934/dcds.2018056
[4]	Chuangxia Huang, Hua Zhang, Lihong Huang. Almost periodicity analysis for a delayed Nicholson's blowflies model with nonlinear density-dependent mortality term. Communications on Pure & Applied Analysis, 2019, 18 (6) : 3337-3349. doi: 10.3934/cpaa.2019150
[5]	Yu Tian, John R. Graef, Lingju Kong, Min Wang. Existence of solutions to a multi-point boundary value problem for a second order differential system via the dual least action principle. Conference Publications, 2013, 2013 (special) : 759-769. doi: 10.3934/proc.2013.2013.759
[6]	Hélène Hibon, Ying Hu, Yiqing Lin, Peng Luo, Falei Wang. Quadratic BSDEs with mean reflection. Mathematical Control & Related Fields, 2018, 8 (3&4) : 721-738. doi: 10.3934/mcrf.2018031
[7]	Peng Sun. Minimality and gluing orbit property. Discrete & Continuous Dynamical Systems - A, 2019, 39 (7) : 4041-4056. doi: 10.3934/dcds.2019162
[8]	Viorel Nitica, Andrei Török. On a semigroup problem. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 2365-2377. doi: 10.3934/dcdss.2019148
[9]	Brandon Seward. Every action of a nonamenable group is the factor of a small action. Journal of Modern Dynamics, 2014, 8 (2) : 251-270. doi: 10.3934/jmd.2014.8.251
[10]	Pedro Duarte, José Pedro GaivÃo, Mohammad Soufi. Hyperbolic billiards on polytopes with contracting reflection laws. Discrete & Continuous Dynamical Systems - A, 2017, 37 (6) : 3079-3109. doi: 10.3934/dcds.2017132
[11]	Terasan Niyomsataya, Ali Miri, Monica Nevins. Decoding affine reflection group codes with trellises. Advances in Mathematics of Communications, 2012, 6 (4) : 385-400. doi: 10.3934/amc.2012.6.385
[12]	Ivan Dynnikov, Alexandra Skripchenko. Minimality of interval exchange transformations with restrictions. Journal of Modern Dynamics, 2017, 11: 219-248. doi: 10.3934/jmd.2017010
[13]	Michael Hutchings. Mean action and the Calabi invariant. Journal of Modern Dynamics, 2016, 10: 511-539. doi: 10.3934/jmd.2016.10.511
[14]	J. W. Neuberger. How to distinguish a local semigroup from a global semigroup. Discrete & Continuous Dynamical Systems - A, 2013, 33 (11&12) : 5293-5303. doi: 10.3934/dcds.2013.33.5293
[15]	Andrzej Biś. Entropies of a semigroup of maps. Discrete & Continuous Dynamical Systems - A, 2004, 11 (2&3) : 639-648. doi: 10.3934/dcds.2004.11.639
[16]	Carlos Cabrera, Peter Makienko, Peter Plaumann. Semigroup representations in holomorphic dynamics. Discrete & Continuous Dynamical Systems - A, 2013, 33 (4) : 1333-1349. doi: 10.3934/dcds.2013.33.1333
[17]	Helmut Kröger. From quantum action to quantum chaos. Conference Publications, 2003, 2003 (Special) : 492-500. doi: 10.3934/proc.2003.2003.492
[18]	Yahong Peng, Yaguang Wang. Reflection of highly oscillatory waves with continuous oscillatory spectra for semilinear hyperbolic systems. Discrete & Continuous Dynamical Systems - A, 2009, 24 (4) : 1293-1306. doi: 10.3934/dcds.2009.24.1293
[19]	Alba Málaga Sabogal, Serge Troubetzkoy. Minimality of the Ehrenfest wind-tree model. Journal of Modern Dynamics, 2016, 10: 209-228. doi: 10.3934/jmd.2016.10.209
[20]	Petr Kůrka. Minimality in iterative systems of Möbius transformations. Conference Publications, 2011, 2011 (Special) : 903-912. doi: 10.3934/proc.2011.2011.903

American Institute of Mathematical Sciences