Lifting the regionally proximal relation and characterizations of distal extensions

Kai Cao; Xiongping Dai

doi:10.3934/dcds.2021186

Previous Article
Global weak solutions to the stochastic Ericksen–Leslie system in dimension two
DCDS Home
This Issue
Next Article
Geodesic planes in geometrically finite manifolds-corrigendum

May 2022, 42(5): 2103-2174. doi: 10.3934/dcds.2021186

Lifting the regionally proximal relation and characterizations of distal extensions

Kai Cao and Xiongping Dai ^,

Department of Mathematics, Nanjing University, Nanjing 210093, China

^*Corresponding author: Xiongping Dai

Received August 2020 Revised August 2021 Published May 2022 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.

Keywords: Distal extension, RP, disjointness, weak-mixing, G-tower.

Mathematics Subject Classification: Primary: 37B05; Secondary: 54H15.

Citation: Kai Cao, Xiongping Dai. Lifting the regionally proximal relation and characterizations of distal extensions. Discrete and Continuous Dynamical Systems, 2022, 42 (5) : 2103-2174. doi: 10.3934/dcds.2021186

References:

[1]	E. Akin and S. Kolyada, Li-Yorke sensitivity, Nonlinearity, 16 (2003), 1421-1433. doi: 10.1088/0951-7715/16/4/313.
[2]	J. Auslander, Endomorphisms of minimal sets, Duke Math. J., 30 (1963), 605-614. doi: 10.1215/S0012-7094-63-03065-5.
[3]	J. Auslander, Regular minimal sets. I, Trans. Amer. Math. Soc., 123 (1966), 469-479. doi: 10.1090/S0002-9947-1966-0193629-4.
[4]	J. Auslander, Homomorphisms of minimal transformation groups, Topology, 9 (1970), 195-203. doi: 10.1016/0040-9383(70)90041-8.
[5]	J. Auslander, Minimal Flows and their Extensions, North-Holland Math. Studies, Vol. 153. North-Holland, Amsterdam, 1988.
[6]	J. Auslander, Minimal flows with a closed proximal cell, Ergod. Th. & Dynam. Sys., 21 (2001), 641-645. doi: 10.1017/S0143385701001316.
[7]	J. Auslander, A group theoretic condition in topological dynamics, Topology Proc., 28 (2004), 327-334.
[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.
[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.
[10]	J. Auslander and E. Glasner, The distal order of a minimal flow, Israel J. Math., 127 (2002), 61-80. doi: 10.1007/BF02784526.
[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.
[12]	J. Auslander and M. Guerin, Regional proximality and the prolongation, Forum Math., 9 (1997), 761-774. doi: 10.1515/form.1997.9.761.
[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.
[14]	I. U. Bronšteǐn, Distal extensions of minimal transformation groups, Sib. Math. J., 11 (1970), 897-911.
[15]	I. U. Bronšteǐn, On the structure of distal extensions and finite-dimensional distal minimal sets, Math. Issled., 6 (1971), 41-62.
[16]	I. U. Bronšteǐn, A theorem on the structure of almost distal expansions of minimal sets, Math. Issled., 6 (1971), 22-32.
[17]	I. U. Bronšteǐn, Stable and equicontinuous extensions of minimal sets, Math. Issled., 8 (1973), 3–11, (Russian).
[18]	I. U. Bronšteǐn, Extensions of Minimal Transformation Groups, Sijthoff & Noordhoff, 1979.
[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.
[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.
[21]	J. de Vries, The Furstenberg structure theorem in topological dynamics, CWI Quarterly, 4 (1991), 27-44.
[22]	J. de Vries, Elements of Topological Dynamics, Kluwer, Dordrecht, 1993. doi: 10.1007/978-94-015-8171-4.
[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.
[24]	R. E. Edwards, Functional Analysis Theory and Applications, Dover Publications, Inc., New York, 1965.
[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.
[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.
[27]	R. Ellis, Distal transformation groups, Pacific J. Math., 8 (1958), 401-405. doi: 10.2140/pjm.1958.8.401.
[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.
[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.
[30]	R. Ellis, Lectures on Topological Dynamics, Benjamin, New York, 1969.
[31]	R. Ellis, The Veech structure theorem, Trans. Amer. Math. Soc., 186 (1973), 203-218. doi: 10.1090/S0002-9947-1973-0350712-1.
[32]	R. Ellis, The Furstenberg structure theorem, Pacific J. Math., 76 (1978), 345-349. doi: 10.2140/pjm.1978.76.345.
[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.
[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.
[35]	H. Furstenberg, The structure of distal flows, Amer. J. Math., 85 (1963), 477-515. doi: 10.2307/2373137.
[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.
[37]	H. Furstenberg, Recurrence in Ergodic Theory and Combinatorial Number Theory, Princeton University Press, Princeton, New Jersey, 1981.
[38]	H. Furstenberg and S. Glasner, On the existence of isometric extensions, Amer. J. Math., 100 (1978), 1185-1212. doi: 10.2307/2373969.
[39]	A. I. Gerko, The structure of distal extensions of minimal transformation semigroups, Mat. Issled., (Kishinev), 7 (1972), 69–86 (Russian).
[40]	A. I. Gerko, Regionally distal extensions and group extensions of semigroups of transformations, Mat. Issled., 10 (1975), 94–106 (Russian).
[41]	A. I. Gerko, RIC-extensions and the structure of extensions of minimal semigroups of transformations, Mat. Issled., No. 77 (1984), 45–60 (Russian).
[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).
[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).
[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.
[45]	A. I. Gerko, Extensions of Topological Transformation Semigroups, State University of Moldova, Publising Center of MolSU, Chisinau, 2001 (Russian).
[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.
[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.
[48]	S. Glasner, Relatively invariant measures, Pacific J. Math., 58 (1975), 393-410. doi: 10.2140/pjm.1975.58.393.
[49]	S. Glasner, Proximal Flows, Lecture Notes in Math., 517, Springer-Verlag, 1976. doi: 10.1007/BFb0080139.
[50]	W. H. Gottschalk, Almost periodic points with respect to transformation semi-groups, Annals of Math., 47 (1946), 762-766. doi: 10.2307/1969233.
[51]	N. Hindman and D. Strauss, Algebra in the Stone-Čech Compactification: Theory and Applications, De Gruyter, Berlin, 1998. doi: 10.1515/9783110809220.
[52]	J. L. Kelley, General Topology, GTM 27, Springer-Verlag, New York Berlin Heidelberg, 1955.
[53]	S. C. Koo, Recursive properties of transformations groups in hyperspaces, Math. Systems Theory, 9 (1975), 75-82. doi: 10.1007/BF01698127.
[54]	D. McMahon, Weak mixing and a note on a structure theorem for minimal transformation groups, Illinois J. Math., 20 (1976), 186-197.
[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.
[56]	D. McMahon and T. S. Wu, On proximal and distal extensions of minimal sets, Bull. Inst. Math. Acad. Sinica, 2 (1974), 93-107.
[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.
[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.
[59]	V. V. Nemytskii and V. V. Stepanov, Qualitative Theory of Differential Equations, Princeton University Press, Princeton, New Jersey, 1960.
[60]	W. Parry and P. Walters, Minimal skew-product homeomorphisms and coalescence, Compos. Math., 22 (1970), 283-288.
[61]	R. Peleg, Weak disjointness of transformation groups, Proc. Amer. Math. Soc., 33 (1972), 165-170. doi: 10.1090/S0002-9939-1972-0298642-2.
[62]	R. R. Phelps, Lectures on Choquet's Theorem, 2$^{nd}$ edition, Lecture Notes in Math., 1757, Springer-Verlag, Berlin Heidelberg New York, 2001. doi: 10.1007/b76887.
[63]	L. Shapiro, Distal and proximal extensions of minimal flows, Math. Systems Theory, 5 (1971), 76-88. doi: 10.1007/BF01691470.
[64]	J. van der Woude, Characterizations of (H)PI extensions, Pacific J. Math., 120 (1985), 453-467. doi: 10.2140/pjm.1985.120.453.
[65]	J. C. S. P. van der Woude, Topological Dynamic, CWI Tracts No 22, Centre for Mathematics and Computer Science (CWI), Amsterdan, 1986.
[66]	W. A. Veech, The equicontinuous structure relation for minimal abelian transformation groups, Amer. J. Math., 90 (1968), 723-732. doi: 10.2307/2373480.
[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.

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.
[2]	J. Auslander, Endomorphisms of minimal sets, Duke Math. J., 30 (1963), 605-614. doi: 10.1215/S0012-7094-63-03065-5.
[3]	J. Auslander, Regular minimal sets. I, Trans. Amer. Math. Soc., 123 (1966), 469-479. doi: 10.1090/S0002-9947-1966-0193629-4.
[4]	J. Auslander, Homomorphisms of minimal transformation groups, Topology, 9 (1970), 195-203. doi: 10.1016/0040-9383(70)90041-8.
[5]	J. Auslander, Minimal Flows and their Extensions, North-Holland Math. Studies, Vol. 153. North-Holland, Amsterdam, 1988.
[6]	J. Auslander, Minimal flows with a closed proximal cell, Ergod. Th. & Dynam. Sys., 21 (2001), 641-645. doi: 10.1017/S0143385701001316.
[7]	J. Auslander, A group theoretic condition in topological dynamics, Topology Proc., 28 (2004), 327-334.
[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.
[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.
[10]	J. Auslander and E. Glasner, The distal order of a minimal flow, Israel J. Math., 127 (2002), 61-80. doi: 10.1007/BF02784526.
[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.
[12]	J. Auslander and M. Guerin, Regional proximality and the prolongation, Forum Math., 9 (1997), 761-774. doi: 10.1515/form.1997.9.761.
[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.
[14]	I. U. Bronšteǐn, Distal extensions of minimal transformation groups, Sib. Math. J., 11 (1970), 897-911.
[15]	I. U. Bronšteǐn, On the structure of distal extensions and finite-dimensional distal minimal sets, Math. Issled., 6 (1971), 41-62.
[16]	I. U. Bronšteǐn, A theorem on the structure of almost distal expansions of minimal sets, Math. Issled., 6 (1971), 22-32.
[17]	I. U. Bronšteǐn, Stable and equicontinuous extensions of minimal sets, Math. Issled., 8 (1973), 3–11, (Russian).
[18]	I. U. Bronšteǐn, Extensions of Minimal Transformation Groups, Sijthoff & Noordhoff, 1979.
[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.
[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.
[21]	J. de Vries, The Furstenberg structure theorem in topological dynamics, CWI Quarterly, 4 (1991), 27-44.
[22]	J. de Vries, Elements of Topological Dynamics, Kluwer, Dordrecht, 1993. doi: 10.1007/978-94-015-8171-4.
[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.
[24]	R. E. Edwards, Functional Analysis Theory and Applications, Dover Publications, Inc., New York, 1965.
[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.
[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.
[27]	R. Ellis, Distal transformation groups, Pacific J. Math., 8 (1958), 401-405. doi: 10.2140/pjm.1958.8.401.
[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.
[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.
[30]	R. Ellis, Lectures on Topological Dynamics, Benjamin, New York, 1969.
[31]	R. Ellis, The Veech structure theorem, Trans. Amer. Math. Soc., 186 (1973), 203-218. doi: 10.1090/S0002-9947-1973-0350712-1.
[32]	R. Ellis, The Furstenberg structure theorem, Pacific J. Math., 76 (1978), 345-349. doi: 10.2140/pjm.1978.76.345.
[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.
[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.
[35]	H. Furstenberg, The structure of distal flows, Amer. J. Math., 85 (1963), 477-515. doi: 10.2307/2373137.
[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.
[37]	H. Furstenberg, Recurrence in Ergodic Theory and Combinatorial Number Theory, Princeton University Press, Princeton, New Jersey, 1981.
[38]	H. Furstenberg and S. Glasner, On the existence of isometric extensions, Amer. J. Math., 100 (1978), 1185-1212. doi: 10.2307/2373969.
[39]	A. I. Gerko, The structure of distal extensions of minimal transformation semigroups, Mat. Issled., (Kishinev), 7 (1972), 69–86 (Russian).
[40]	A. I. Gerko, Regionally distal extensions and group extensions of semigroups of transformations, Mat. Issled., 10 (1975), 94–106 (Russian).
[41]	A. I. Gerko, RIC-extensions and the structure of extensions of minimal semigroups of transformations, Mat. Issled., No. 77 (1984), 45–60 (Russian).
[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).
[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).
[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.
[45]	A. I. Gerko, Extensions of Topological Transformation Semigroups, State University of Moldova, Publising Center of MolSU, Chisinau, 2001 (Russian).
[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.
[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.
[48]	S. Glasner, Relatively invariant measures, Pacific J. Math., 58 (1975), 393-410. doi: 10.2140/pjm.1975.58.393.
[49]	S. Glasner, Proximal Flows, Lecture Notes in Math., 517, Springer-Verlag, 1976. doi: 10.1007/BFb0080139.
[50]	W. H. Gottschalk, Almost periodic points with respect to transformation semi-groups, Annals of Math., 47 (1946), 762-766. doi: 10.2307/1969233.
[51]	N. Hindman and D. Strauss, Algebra in the Stone-Čech Compactification: Theory and Applications, De Gruyter, Berlin, 1998. doi: 10.1515/9783110809220.
[52]	J. L. Kelley, General Topology, GTM 27, Springer-Verlag, New York Berlin Heidelberg, 1955.
[53]	S. C. Koo, Recursive properties of transformations groups in hyperspaces, Math. Systems Theory, 9 (1975), 75-82. doi: 10.1007/BF01698127.
[54]	D. McMahon, Weak mixing and a note on a structure theorem for minimal transformation groups, Illinois J. Math., 20 (1976), 186-197.
[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.
[56]	D. McMahon and T. S. Wu, On proximal and distal extensions of minimal sets, Bull. Inst. Math. Acad. Sinica, 2 (1974), 93-107.
[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.
[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.
[59]	V. V. Nemytskii and V. V. Stepanov, Qualitative Theory of Differential Equations, Princeton University Press, Princeton, New Jersey, 1960.
[60]	W. Parry and P. Walters, Minimal skew-product homeomorphisms and coalescence, Compos. Math., 22 (1970), 283-288.
[61]	R. Peleg, Weak disjointness of transformation groups, Proc. Amer. Math. Soc., 33 (1972), 165-170. doi: 10.1090/S0002-9939-1972-0298642-2.
[62]	R. R. Phelps, Lectures on Choquet's Theorem, 2$^{nd}$ edition, Lecture Notes in Math., 1757, Springer-Verlag, Berlin Heidelberg New York, 2001. doi: 10.1007/b76887.
[63]	L. Shapiro, Distal and proximal extensions of minimal flows, Math. Systems Theory, 5 (1971), 76-88. doi: 10.1007/BF01691470.
[64]	J. van der Woude, Characterizations of (H)PI extensions, Pacific J. Math., 120 (1985), 453-467. doi: 10.2140/pjm.1985.120.453.
[65]	J. C. S. P. van der Woude, Topological Dynamic, CWI Tracts No 22, Centre for Mathematics and Computer Science (CWI), Amsterdan, 1986.
[66]	W. A. Veech, The equicontinuous structure relation for minimal abelian transformation groups, Amer. J. Math., 90 (1968), 723-732. doi: 10.2307/2373480.
[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.

[1]	Eli Glasner, Benjamin Weiss. A generic distal tower of arbitrary countable height over an arbitrary infinite ergodic system. Journal of Modern Dynamics, 2021, 17: 435-463. doi: 10.3934/jmd.2021015
[2]	Mariusz Lemańczyk. Ergodicity, mixing, Ratner's properties and disjointness for classical flows: On the research of Corinna Ulcigrai. Journal of Modern Dynamics, 2022, 18: 103-130. doi: 10.3934/jmd.2022005
[3]	Jean René Chazottes, F. Durand. Local rates of Poincaré recurrence for rotations and weak mixing. Discrete and Continuous Dynamical Systems, 2005, 12 (1) : 175-183. doi: 10.3934/dcds.2005.12.175
[4]	Oliver Knill. Singular continuous spectrum and quantitative rates of weak mixing. Discrete and Continuous Dynamical Systems, 1998, 4 (1) : 33-42. doi: 10.3934/dcds.1998.4.33
[5]	Ethan M. Ackelsberg. Rigidity, weak mixing, and recurrence in abelian groups. Discrete and Continuous Dynamical Systems, 2022, 42 (4) : 1669-1705. doi: 10.3934/dcds.2021168
[6]	Anthony Quas, Terry Soo. Weak mixing suspension flows over shifts of finite type are universal. Journal of Modern Dynamics, 2012, 6 (4) : 427-449. doi: 10.3934/jmd.2012.6.427
[7]	Corinna Ulcigrai. Weak mixing for logarithmic flows over interval exchange transformations. Journal of Modern Dynamics, 2009, 3 (1) : 35-49. doi: 10.3934/jmd.2009.3.35
[8]	Roberto De Leo, James A. Yorke. The graph of the logistic map is a tower. Discrete and Continuous Dynamical Systems, 2021, 41 (11) : 5243-5269. doi: 10.3934/dcds.2021075
[9]	Jie Li, Kesong Yan, Xiangdong Ye. Recurrence properties and disjointness on the induced spaces. Discrete and Continuous Dynamical Systems, 2015, 35 (3) : 1059-1073. doi: 10.3934/dcds.2015.35.1059
[10]	Krzysztof Frączek, Leonid Polterovich. Growth and mixing. Journal of Modern Dynamics, 2008, 2 (2) : 315-338. doi: 10.3934/jmd.2008.2.315
[11]	Ingenuin Gasser. Modelling and simulation of a solar updraft tower. Kinetic and Related Models, 2009, 2 (1) : 191-204. doi: 10.3934/krm.2009.2.191
[12]	Gernot Greschonig. Real cocycles of point-distal minimal flows. Conference Publications, 2015, 2015 (special) : 540-548. doi: 10.3934/proc.2015.0540
[13]	Piotr Oprocha. Double minimality, entropy and disjointness with all minimal systems. Discrete and Continuous Dynamical Systems, 2019, 39 (1) : 263-275. doi: 10.3934/dcds.2019011
[14]	Philipp Kunde. Smooth diffeomorphisms with homogeneous spectrum and disjointness of convolutions. Journal of Modern Dynamics, 2016, 10: 439-481. doi: 10.3934/jmd.2016.10.439
[15]	Wen Huang, Jianya Liu, Ke Wang. Möbius disjointness for skew products on a circle and a nilmanifold. Discrete and Continuous Dynamical Systems, 2021, 41 (8) : 3531-3553. doi: 10.3934/dcds.2021006
[16]	David Ginzburg and Joseph Hundley. A new tower of Rankin-Selberg integrals. Electronic Research Announcements, 2006, 12: 56-62.
[17]	Jacek Serafin. A faithful symbolic extension. Communications on Pure and Applied Analysis, 2012, 11 (3) : 1051-1062. doi: 10.3934/cpaa.2012.11.1051
[18]	Hans F. Weinberger, Xiao-Qiang Zhao. An extension of the formula for spreading speeds. Mathematical Biosciences & Engineering, 2010, 7 (1) : 187-194. doi: 10.3934/mbe.2010.7.187
[19]	Augusto Visintin. An extension of the Fitzpatrick theory. Communications on Pure and Applied Analysis, 2014, 13 (5) : 2039-2058. doi: 10.3934/cpaa.2014.13.2039
[20]	Asaf Katz. On mixing and sparse ergodic theorems. Journal of Modern Dynamics, 2021, 17: 1-32. doi: 10.3934/jmd.2021001

2020 Impact Factor: 1.392

Tools

Metrics

Tools

Article outline

Figures and Tables

2020 Impact Factor: 1.392

Tools

Figures and Tables

2020 Impact Factor: 1.392

American Institute of Mathematical Sciences

Lifting the regionally proximal relation and characterizations of distal extensions

References:

References:

Tools

Metrics

Other articles
by authors

Tools

Tools

Tools

Metrics

Other articles
by authors

American Institute of Mathematical Sciences

Lifting the regionally proximal relation and characterizations of distal extensions

References:

References:

Tools

Metrics

Other articlesby authors

Tools

Tools

Tools

Metrics

Other articlesby authors

Other articles
by authors

Other articles
by authors