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Geodesic planes in geometrically finite manifolds-corrigendum
Lifting the regionally proximal relation and characterizations of distal extensions
Department of Mathematics, Nanjing University, Nanjing 210093, China |
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.
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. |
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W. A. Veech,
The equicontinuous structure relation for minimal abelian transformation groups, Amer. J. Math., 90 (1968), 723-732.
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[67] |
W. A. Veech,
Point-distal flows, Amer. J. Math., 92 (1970), 205-242.
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[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. |
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