# American Institute of Mathematical Sciences

doi: 10.3934/dcds.2021186
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## Lifting the regionally proximal relation and characterizations of distal extensions

 Department of Mathematics, Nanjing University, Nanjing 210093, China

*Corresponding author: Xiongping Dai

Received  August 2020 Revised  August 2021 Early access December 2021

Fund Project: The second author is supported by National Natural Science Foundation of China (Grant No. 11790274) and PAPD of Jiangsu Higher Education Institutions

We consider a commutative diagram (CD) of flows with discrete phase group $T$ and extensions as follows:

By ${\text{RP}}_{\!\phi}$ and ${\text{RP}}_{\!\psi}$ we denote the relativized regionally proximal relations in $X$ and $Y$, respectively. We mainly prove, among other things, the following:

1. If $X$ is topologically transitive $\phi$-distal, then $X$ is minimal.

2. $(\pi\times\pi) {\text{RP}}_{\!\phi} = {\text{RP}}_{\!\psi}$.

3. If $Y$ is locally $\psi$-Bronstein, then ${\text{RP}}_{\!\psi}\circ {\text{P}}_{\!\psi}$ is an equivalence relation, and, $\bar{y}\in {\text{RP}}_{\!\psi}$ whenever $\bar{y}\in {\text{RP}}_{\!\psi}\circ {\text{RP}}_{\!\psi}$ is almost periodic.

4. If $Y_d$ is the maximal distal extension of $Z$ below $Y$ and $Z^d$ is the universal minimal distal extension of $Z$, then $Y\perp_{Y_d}Z^d$.

5. (a) $Y$ is locally $\psi$-Bronstein iff $F^d<F^\prime A$ where $A,F,F^d$ are respectively the Ellis groups of $Y,Z, Z^d$. (b) If $Y$ is locally $\psi$-Bronstein and $K$ a $\tau$-closed group with $F^\prime A<K<F$, then there is a unique $Y_{\!K}$ which is $\psi_{\!K}^\prime$-equicontinuous and has the Ellis group $K$.

We also prove the above theorems in the case $T$ is a semigroup. Moreover, we show the following in minimal semiflows:

6. $Y$ is $\psi$-distal iff $\psi$ has a DE-tower iff there is a least group-like extension $X$ via $\phi$ (i.e., $\phi^{-1}\phi x = {\text{Aut}}_\phi(T,X)x$ for all $x\in X$).

7. $\psi$ is group-like iff $\psi$ has a G-tower that consists of group extensions and inverse limits.

Citation: Kai Cao, Xiongping Dai. Lifting the regionally proximal relation and characterizations of distal extensions. Discrete & Continuous Dynamical Systems, doi: 10.3934/dcds.2021186
##### References:
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Math., 58 (1975), 393-410.  doi: 10.2140/pjm.1975.58.393.  Google Scholar [49] S. Glasner, Proximal Flows, Lecture Notes in Math., 517, Springer-Verlag, 1976. doi: 10.1007/BFb0080139.  Google Scholar [50] 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 [51] N. Hindman and D. Strauss, Algebra in the Stone-Čech Compactification: Theory and Applications, De Gruyter, Berlin, 1998. doi: 10.1515/9783110809220.  Google Scholar [52] J. L. Kelley, General Topology, GTM 27, Springer-Verlag, New York Berlin Heidelberg, 1955.  Google Scholar [53] S. C. Koo, Recursive properties of transformations groups in hyperspaces, Math. Systems Theory, 9 (1975), 75-82.  doi: 10.1007/BF01698127.  Google Scholar [54] D. McMahon, Weak mixing and a note on a structure theorem for minimal transformation groups, Illinois J. Math., 20 (1976), 186-197.   Google Scholar [55] D. C. 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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

show all references

##### References:
 [1] E. Akin and S. Kolyada, Li-Yorke sensitivity, Nonlinearity, 16 (2003), 1421-1433.  doi: 10.1088/0951-7715/16/4/313.  Google Scholar [2] J. Auslander, Endomorphisms of minimal sets, Duke Math. J., 30 (1963), 605-614.  doi: 10.1215/S0012-7094-63-03065-5.  Google Scholar [3] J. Auslander, Regular minimal sets. I, Trans. Amer. Math. Soc., 123 (1966), 469-479.  doi: 10.1090/S0002-9947-1966-0193629-4.  Google Scholar [4] J. Auslander, Homomorphisms of minimal transformation groups, Topology, 9 (1970), 195-203.  doi: 10.1016/0040-9383(70)90041-8.  Google Scholar [5] J. Auslander, Minimal Flows and their Extensions, North-Holland Math. Studies, Vol. 153. North-Holland, Amsterdam, 1988.  Google Scholar [6] J. Auslander, Minimal flows with a closed proximal cell, Ergod. Th. & Dynam. Sys., 21 (2001), 641-645.  doi: 10.1017/S0143385701001316.  Google Scholar [7] J. Auslander, A group theoretic condition in topological dynamics, Topology Proc., 28 (2004), 327-334.   Google Scholar [8] J. Auslander and X. Dai, Minimality, distality and equicontinuity for semigroup actions on compact Hausdorff spaces, Discrete Contin. Dyn. Syst., 39 (2019), 4647-4711.  doi: 10.3934/dcds.2019190.  Google Scholar [9] J. Auslander, D. B. Ellis and R. Ellis, The regionally proximal relation, Trans. Amer. Math. Soc., 347 (1995), 2139-2146.  doi: 10.1090/S0002-9947-1995-1285970-7.  Google Scholar [10] J. Auslander and E. Glasner, The distal order of a minimal flow, Israel J. Math., 127 (2002), 61-80.  doi: 10.1007/BF02784526.  Google Scholar [11] J. Auslander and E. Glasner, On the virtual automorphism group of a minimal flow, Ergod. Th. & Dynam. Sys., 41 (2021), 1281-1295.  doi: 10.1017/etds.2020.8.  Google Scholar [12] J. Auslander and M. Guerin, Regional proximality and the prolongation, Forum Math., 9 (1997), 761-774.  doi: 10.1515/form.1997.9.761.  Google Scholar [13] J. Auslander, D. McMahon, J. van der Woude and T. S. Wu, Weak disjointness and the equicontinuous structure relation, Ergod. Th. & Dynam. Sys., 4 (1984), 323-351.  doi: 10.1017/S0143385700002492.  Google Scholar [14] I. U. Bronšteǐn, Distal extensions of minimal transformation groups, Sib. Math. J., 11 (1970), 897-911.   Google Scholar [15] I. U. Bronšteǐn, On the structure of distal extensions and finite-dimensional distal minimal sets, Math. Issled., 6 (1971), 41-62.   Google Scholar [16] I. U. Bronšteǐn, A theorem on the structure of almost distal expansions of minimal sets, Math. Issled., 6 (1971), 22-32.   Google Scholar [17] I. U. Bronšteǐn, Stable and equicontinuous extensions of minimal sets, Math. Issled., 8 (1973), 3–11, (Russian).  Google Scholar [18] I. U. Bronšteǐn, Extensions of Minimal Transformation Groups, Sijthoff & Noordhoff, 1979. Google Scholar [19] I. U. Bronšteǐn and A. I. Gerko, The embedding of certain topological transformation semigroups into topological transformation groups, Bul. Akad. Štiince RSS Moldoven, (1970), 18–24.  Google Scholar [20] X. Dai, Almost automorphy and Veech's relations of dynamics on compact $T_2$-spaces, J. Differential Equations, 269 (2020), 2580-2626.  doi: 10.1016/j.jde.2020.02.005.  Google Scholar [21] J. de Vries, The Furstenberg structure theorem in topological dynamics, CWI Quarterly, 4 (1991), 27-44.   Google Scholar [22] J. de Vries, Elements of Topological Dynamics, Kluwer, Dordrecht, 1993. doi: 10.1007/978-94-015-8171-4.  Google Scholar [23] T. Downarowicz, The royal couple conceals their mutual relationship: A noncoalescent Toeplitz flow, Israel J. Math., 97 (1997), 239-251.  doi: 10.1007/BF02774039.  Google Scholar [24] R. E. Edwards, Functional Analysis Theory and Applications, Dover Publications, Inc., New York, 1965.  Google Scholar [25] D. B. 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 [26] D. B. 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 [27] R. Ellis, Distal transformation groups, Pacific J. Math., 8 (1958), 401-405.  doi: 10.2140/pjm.1958.8.401.  Google Scholar [28] R. Ellis, Group-like extensions of minimal sets, Trans. Amer. Math. Soc., 127 (1967), 125-135.  doi: 10.1090/S0002-9947-1967-0221492-2.  Google Scholar [29] R. Ellis, The structure of group-like extensions of minimal sets, Trans. Amer. Math. Soc., 134 (1968), 261-287.  doi: 10.1090/S0002-9947-1968-0238293-2.  Google Scholar [30] R. Ellis, Lectures on Topological Dynamics, Benjamin, New York, 1969.  Google Scholar [31] R. Ellis, The Veech structure theorem, Trans. Amer. Math. Soc., 186 (1973), 203-218.  doi: 10.1090/S0002-9947-1973-0350712-1.  Google Scholar [32] R. Ellis, The Furstenberg structure theorem, Pacific J. Math., 76 (1978), 345-349.  doi: 10.2140/pjm.1978.76.345.  Google Scholar [33] R. Ellis, S. Glasner and L. Shapiro, Proximal-isometric $(\mathscr{P}\!\!\mathscr{I})$ flows, Adv. in Math., 17 (1975), 213-260.  doi: 10.1016/0001-8708(75)90093-6.  Google Scholar [34] R. Ellis and W. H. Gottschalk, Homomorphisms of transformation groups, Trans. Amer. Math. Soc., 94 (1960), 258-271.  doi: 10.1090/S0002-9947-1960-0123635-1.  Google Scholar [35] H. Furstenberg, The structure of distal flows, Amer. J. Math., 85 (1963), 477-515.  doi: 10.2307/2373137.  Google Scholar [36] H. Furstenberg, Disjointness in ergodic theory, minimal sets, and a problem in Diophantine approximation, Math. Systems Theory, 1 (1967), 1-49.  doi: 10.1007/BF01692494.  Google Scholar [37] H. Furstenberg, Recurrence in Ergodic Theory and Combinatorial Number Theory, Princeton University Press, Princeton, New Jersey, 1981.   Google Scholar [38] H. Furstenberg and S. Glasner, On the existence of isometric extensions, Amer. J. Math., 100 (1978), 1185-1212.  doi: 10.2307/2373969.  Google Scholar [39] A. I. Gerko, The structure of distal extensions of minimal transformation semigroups, Mat. Issled., (Kishinev), 7 (1972), 69–86 (Russian).  Google Scholar [40] A. I. Gerko, Regionally distal extensions and group extensions of semigroups of transformations, Mat. Issled., 10 (1975), 94–106 (Russian).  Google Scholar [41] A. I. Gerko, RIC-extensions and the structure of extensions of minimal semigroups of transformations, Mat. Issled., No. 77 (1984), 45–60 (Russian).  Google Scholar [42] A. I. Gerko, $\tau(A)$-topology and certain properties of groups associated with topological transformation semigroups, Izv. Akad. Nauk Moldovy, (1991), 26–31 (Russian).  Google Scholar [43] A. I. Gerko, On the structure of $n$-fans of minimal topological transformation semigroups, Scientific Annals, Faculty of Mathematics and Informatics, State Univ. of Moldova, 1 (1999), 50–74 (Russian). Google Scholar [44] A. I. Gerko, On some disjointness classes of extensions of minimal topological transformation semigroups, Ukraïn Mat. Zh., 52 (2000), 1335–1344. doi: 10.1023/A:1010496900139.  Google Scholar [45] A. I. Gerko, Extensions of Topological Transformation Semigroups, State University of Moldova, Publising Center of MolSU, Chisinau, 2001 (Russian). Google Scholar [46] E. Glasner, Regular PI metric flows are equicontinuous, Proc. Amer. Math. Soc., 114 (1992), 269-277.  doi: 10.1090/S0002-9939-1992-1070517-3.  Google Scholar [47] E. Glasner, The structure of tame minimal dynamical systems for general groups, Invent. Math., 211 (2018), 213-244.  doi: 10.1007/s00222-017-0747-z.  Google Scholar [48] S. Glasner, Relatively invariant measures, Pacific J. Math., 58 (1975), 393-410.  doi: 10.2140/pjm.1975.58.393.  Google Scholar [49] S. Glasner, Proximal Flows, Lecture Notes in Math., 517, Springer-Verlag, 1976. doi: 10.1007/BFb0080139.  Google Scholar [50] 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 [51] N. Hindman and D. Strauss, Algebra in the Stone-Čech Compactification: Theory and Applications, De Gruyter, Berlin, 1998. doi: 10.1515/9783110809220.  Google Scholar [52] J. L. Kelley, General Topology, GTM 27, Springer-Verlag, New York Berlin Heidelberg, 1955.  Google Scholar [53] S. C. Koo, Recursive properties of transformations groups in hyperspaces, Math. Systems Theory, 9 (1975), 75-82.  doi: 10.1007/BF01698127.  Google Scholar [54] D. McMahon, Weak mixing and a note on a structure theorem for minimal transformation groups, Illinois J. Math., 20 (1976), 186-197.   Google Scholar [55] D. C. McMahon, Relativized weak disjointness and relatively invariant measures, Trans. Amer. Math. Soc., 236 (1978), 225-237.  doi: 10.1090/S0002-9947-1978-0467704-9.  Google Scholar [56] D. McMahon and T. S. Wu, On proximal and distal extensions of minimal sets, Bull. Inst. Math. Acad. Sinica, 2 (1974), 93-107.   Google Scholar [57] D. McMahon and T. S. Wu, Distal homomorphisms of nonmetric minimal flows, Proc. Amer. Math. Soc., 82 (1981), 283-287.  doi: 10.1090/S0002-9939-1981-0609668-6.  Google Scholar [58] W. B. Moors and I. Namioka, Furstenberg's structure theorem via CHART groups, Ergod. Th. & Dynam. Sys., 33 (2013), 954-968.  doi: 10.1017/S0143385712000089.  Google Scholar [59] V. V. Nemytskii and V. V. Stepanov, Qualitative Theory of Differential Equations, Princeton University Press, Princeton, New Jersey, 1960.   Google Scholar [60] W. Parry and P. Walters, Minimal skew-product homeomorphisms and coalescence, Compos. Math., 22 (1970), 283-288.   Google Scholar [61] R. 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