December  2020, 40(12): 6855-6875. doi: 10.3934/dcds.2020132

Maximal equicontinuous generic factors and weak model sets

Department Mathematik, Universität Erlangen-Nürnberg, Cauerstr. 11, 91058 Erlangen, Germany

Received  May 2019 Revised  December 2019 Published  December 2020 Early access  February 2020

The orbit closures of regular model sets generated from a cut-and-project scheme given by a co-compact lattice $ {\mathcal L}\subset G\times H $ and compact and aperiodic window $ W\subseteq H $, have the maximal equicontinuous factor (MEF) $ (G\times H)/ {\mathcal L} $, if the window is toplogically regular. This picture breaks down, when the window has empty interior, in which case the MEF is trivial, although $ (G\times H)/ {\mathcal L} $ continues to be the Kronecker factor for the Mirsky measure. As this happens for many interesting examples like the square-free numbers or the visible lattice points, a weaker concept of topological factors is needed, like that of generic factors [24]. For topological dynamical systems that possess a finite invariant measure with full support ($ E $-systems) we prove the existence of a maximal equicontinuous generic factor (MEGF) and characterize it in terms of the regional proximal relation. This part of the paper profits strongly from previous work by McMahon [33] and Auslander [2]. In Sections 3 and 4 we determine the MEGF of orbit closures of weak model sets and use this result for an alternative proof (of a generalization) of the fact [34] that the centralizer of any $ {\mathcal B} $-free dynamical system of Erdős type is trivial.

Citation: Gerhard Keller. Maximal equicontinuous generic factors and weak model sets. Discrete and Continuous Dynamical Systems, 2020, 40 (12) : 6855-6875. doi: 10.3934/dcds.2020132
References:
[1]

J.-B. Aujogue, M. Barge, J. Kellendonk and D. Lenz, Equicontinuous factors, proximality and ellis semigroup for delone sets, in Mathematics of Aperiodic Order (eds. Author 3, Author 4 and J. Savinien), Birkhäuser, 309 (2015), 137–194.

[2]

J. Auslander, Minimal Flows and their Extensions, vol. 153 of North Holland Mathematics Studies, 1988.

[3]

M. BaakeD. Damanik and U. Grimm, What is aperiodic order?, Notices of the American Mathematical Society, 63 (2016), 647-650.  doi: 10.1090/noti1394.

[4]

M. Baake and U. Grimm, Aperiodic Order. Vol. 1: A Mathematical Invitation, vol. 149 of Encyclopedia of Mathematics and its Applications, Cambridge University Press, 2013. doi: 10.1007/s00283-014-9487-8.

[5]

M. Baake, U. Grimm and R. V. Moody, What is aperiodic order?, Notices of the AMS, , 63 (2016), 647–650, arXiv: math/0203252. doi: 10.1090/noti1394.

[6]

M. Baake and C. Huck, Ergodic properties of visible lattice points, Proc. Steklov Inst. Math., 288 (2015), 165-188.  doi: 10.1134/S0081543815010137.

[7]

M. Baake and D. Lenz, Dynamical systems on translation bounded measures: Pure point dynamical and diffraction spectra, Ergodic Theory and Dynamical Systems, 24 (2004), 1867-1893.  doi: 10.1017/S0143385704000318.

[8]

F. BlanchardB. Host and A. Maass, Topological complexity, Ergodic Theory and Dynamical Systems, 20 (2000), 641-662.  doi: 10.1017/S0143385700000341.

[9]

G. CairnsA. Kolganova and A. Nielsen, Topological transitivity and mixing notions for group actions, Rocky Mountain Journal of Mathematics, 37 (2007), 371-397.  doi: 10.1216/rmjm/1181068757.

[10]

C. Castaing and M. Valadier, Convex Analysis and Measurable Multifunctions, vol. 580 of Lecture Notes in Mathematics, Springer, 1977. doi: 10.1007/BFb0087685.

[11]

F. Cellarosi and Y. G. Sinai, Ergodic properties of square-free numbers, J. Eur. Math. Soc., 15 (2013), 1343-1374.  doi: 10.4171/JEMS/394.

[12]

H. Davenport and P. Erdős, On sequences of positive integers, Acta Arithmetica, 2 (1936), 147-151.  doi: 10.4064/aa-2-1-147-151.

[13]

H. Davenport and P. Erdős, On sequences of positive integers, J. Indian Math. Soc. (N.S.), 15 (1951), 19-24. 

[14]

T. Downarowicz, Weakly almost periodic flows and hidden eigenvalues,, in Topological Dynamics and Applications (eds. M. Nerurkar, D. Dokken and D. Ellis), vol. 215 of AMS Contemporary Math. Series, 101–120. Amer. Math. Soc., Providence, RI, 1998. doi: 10.1090/conm/215/02933.

[15]

A. Dymek, Automorphisms of Toeplitz $ {\mathcal B}$-free systems, Bull. Pol. Acad. Sc. Math., 65 (2017), 139-152.  doi: 10.4064/ba8115-10-2017.

[16]

A. Dymek, S. Kasjan and G. Keller, Automorphisms of $ {\mathcal B}$-free Toeplitz systems, in preparation.

[17]

A. DymekS. KasjanJ. Kułaga-Przymus and M. Lemańczyk, ${ {\mathcal B}}$-free sets and dynamics, Trans. Amer. Math. Soc., 370 (2018), 5425-5489.  doi: 10.1090/tran/7132.

[18]

H. Furstenberg, Disjointness in ergodic theory, minimal sets, and a problem in diophantine approximation, Mathematical Systems Theory, 1 (1967), 1-49.  doi: 10.1007/BF01692494.

[19]

H. FurstenbergY. Peres and B. Weiss, Perfect filtering and double disjointness, Annales de l'I.H.P., section B, 31 (1995), 453-465. 

[20]

E. Glasner and B. Weiss, Sensitive dependence on initial conditions, Nonlinearity, 6 (1993), 1067-1075.  doi: 10.1088/0951-7715/6/6/014.

[21]

E. Glasner and B. Weiss, Locally equicontinuous dynamical systems, Colloquium Mathematicum, 84/85 (2000), 345-361.  doi: 10.4064/cm-84/85-2-345-361.

[22]

P. GlendinningT. Jäger and G. Keller, How chaotic are strange non-chaotic attractors?, Nonlinearity, 19 (2006), 2005-2022.  doi: 10.1088/0951-7715/19/9/001.

[23]

R. R. Hall, Sets of Multiples, vol. 118 of Cambridge Tracts in Mathematics, Cambridge University Press, 1996. doi: 10.1017/CBO9780511566011.

[24]

W. Huang and X. Ye, Generic eigenvalues, generic factors and weak disjointness, Dynamical Systems and Group Actions, 119–142, Contemp. Math., 567, Amer. Math. Soc., Providence, RI, 2012. doi: 10.1090/conm/567.

[25]

O. Kallenberg, Foundations of Modern Probability, $2^nd$ edition, Springer, New York, 2002. doi: 10.1007/978-1-4757-4015-8.

[26]

S. KasjanG. Keller and M. Lemańczyk, Dynamics of $\mathcal B$-free sets: A view through the window, Int. Math. Res. Notices, 2019 (2019), 2690-2734.  doi: 10.1093/imrn/rnx196.

[27]

G. Keller, Tautness for sets of multiples and applications to $ {\mathcal B}$-free dynamics, Studia Mathematica, 247 (2019), 205-216.  doi: 10.4064/sm180305-9-4.

[28]

G. Keller and C. Richard, Dynamics on the graph of the torus parametrisation, Ergod. Th. & Dynam. Sys., 38 (2018), 1048-1085.  doi: 10.1017/etds.2016.53.

[29]

G. Keller and C. Richard, Periods and factors of weak model sets, Israel J. Math., 229 (2019), 85-132.  doi: 10.1007/s11856-018-1788-8.

[30]

H. B. Keynes and J. B. Robertson, Eigenvalue theorems in topological transformation groups, Transactions of the American Mathematical Society, 139 (1969), 359-369.  doi: 10.1090/S0002-9947-1969-0237748-5.

[31]

A. Kharazishvili, Topics in Measure Theory and Real Analysis, Atlantis Press / World Scientific, Amsterdam - Paris, 2009. doi: 10.2991/978-94-91216-36-7.

[32]

J. Kułaga-PrzymusM. Lemańczyk and B. Weiss, On invariant measures for ${ {\mathcal B}}$-free systems, Proceedings of the London Mathematical Society, 110 (2015), 1435-1474.  doi: 10.1112/plms/pdv017.

[33]

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.

[34]

M. K. Mentzen, Automorphisms of subshifts defined by $ {\mathcal B}$-free sets of integers., Colloquium Mathematicum, 147 (2017), 87-94.  doi: 10.4064/cm6927-5-2016.

[35]

R. V. Moody, Uniform distribution in model sets, Canad. Math. Bull., 45 (2002), 123-130.  doi: 10.4153/CMB-2002-015-3.

[36]

J. R. Munkres, Topology, Prentice Hall, $2^nd$ edition, 2000.

[37]

R. Peckner, Uniqueness of the measure of maximal entropy for the squarefree flow, Israel J. Math., 210 (2015), 335-357.  doi: 10.1007/s11856-015-1255-8.

show all references

References:
[1]

J.-B. Aujogue, M. Barge, J. Kellendonk and D. Lenz, Equicontinuous factors, proximality and ellis semigroup for delone sets, in Mathematics of Aperiodic Order (eds. Author 3, Author 4 and J. Savinien), Birkhäuser, 309 (2015), 137–194.

[2]

J. Auslander, Minimal Flows and their Extensions, vol. 153 of North Holland Mathematics Studies, 1988.

[3]

M. BaakeD. Damanik and U. Grimm, What is aperiodic order?, Notices of the American Mathematical Society, 63 (2016), 647-650.  doi: 10.1090/noti1394.

[4]

M. Baake and U. Grimm, Aperiodic Order. Vol. 1: A Mathematical Invitation, vol. 149 of Encyclopedia of Mathematics and its Applications, Cambridge University Press, 2013. doi: 10.1007/s00283-014-9487-8.

[5]

M. Baake, U. Grimm and R. V. Moody, What is aperiodic order?, Notices of the AMS, , 63 (2016), 647–650, arXiv: math/0203252. doi: 10.1090/noti1394.

[6]

M. Baake and C. Huck, Ergodic properties of visible lattice points, Proc. Steklov Inst. Math., 288 (2015), 165-188.  doi: 10.1134/S0081543815010137.

[7]

M. Baake and D. Lenz, Dynamical systems on translation bounded measures: Pure point dynamical and diffraction spectra, Ergodic Theory and Dynamical Systems, 24 (2004), 1867-1893.  doi: 10.1017/S0143385704000318.

[8]

F. BlanchardB. Host and A. Maass, Topological complexity, Ergodic Theory and Dynamical Systems, 20 (2000), 641-662.  doi: 10.1017/S0143385700000341.

[9]

G. CairnsA. Kolganova and A. Nielsen, Topological transitivity and mixing notions for group actions, Rocky Mountain Journal of Mathematics, 37 (2007), 371-397.  doi: 10.1216/rmjm/1181068757.

[10]

C. Castaing and M. Valadier, Convex Analysis and Measurable Multifunctions, vol. 580 of Lecture Notes in Mathematics, Springer, 1977. doi: 10.1007/BFb0087685.

[11]

F. Cellarosi and Y. G. Sinai, Ergodic properties of square-free numbers, J. Eur. Math. Soc., 15 (2013), 1343-1374.  doi: 10.4171/JEMS/394.

[12]

H. Davenport and P. Erdős, On sequences of positive integers, Acta Arithmetica, 2 (1936), 147-151.  doi: 10.4064/aa-2-1-147-151.

[13]

H. Davenport and P. Erdős, On sequences of positive integers, J. Indian Math. Soc. (N.S.), 15 (1951), 19-24. 

[14]

T. Downarowicz, Weakly almost periodic flows and hidden eigenvalues,, in Topological Dynamics and Applications (eds. M. Nerurkar, D. Dokken and D. Ellis), vol. 215 of AMS Contemporary Math. Series, 101–120. Amer. Math. Soc., Providence, RI, 1998. doi: 10.1090/conm/215/02933.

[15]

A. Dymek, Automorphisms of Toeplitz $ {\mathcal B}$-free systems, Bull. Pol. Acad. Sc. Math., 65 (2017), 139-152.  doi: 10.4064/ba8115-10-2017.

[16]

A. Dymek, S. Kasjan and G. Keller, Automorphisms of $ {\mathcal B}$-free Toeplitz systems, in preparation.

[17]

A. DymekS. KasjanJ. Kułaga-Przymus and M. Lemańczyk, ${ {\mathcal B}}$-free sets and dynamics, Trans. Amer. Math. Soc., 370 (2018), 5425-5489.  doi: 10.1090/tran/7132.

[18]

H. Furstenberg, Disjointness in ergodic theory, minimal sets, and a problem in diophantine approximation, Mathematical Systems Theory, 1 (1967), 1-49.  doi: 10.1007/BF01692494.

[19]

H. FurstenbergY. Peres and B. Weiss, Perfect filtering and double disjointness, Annales de l'I.H.P., section B, 31 (1995), 453-465. 

[20]

E. Glasner and B. Weiss, Sensitive dependence on initial conditions, Nonlinearity, 6 (1993), 1067-1075.  doi: 10.1088/0951-7715/6/6/014.

[21]

E. Glasner and B. Weiss, Locally equicontinuous dynamical systems, Colloquium Mathematicum, 84/85 (2000), 345-361.  doi: 10.4064/cm-84/85-2-345-361.

[22]

P. GlendinningT. Jäger and G. Keller, How chaotic are strange non-chaotic attractors?, Nonlinearity, 19 (2006), 2005-2022.  doi: 10.1088/0951-7715/19/9/001.

[23]

R. R. Hall, Sets of Multiples, vol. 118 of Cambridge Tracts in Mathematics, Cambridge University Press, 1996. doi: 10.1017/CBO9780511566011.

[24]

W. Huang and X. Ye, Generic eigenvalues, generic factors and weak disjointness, Dynamical Systems and Group Actions, 119–142, Contemp. Math., 567, Amer. Math. Soc., Providence, RI, 2012. doi: 10.1090/conm/567.

[25]

O. Kallenberg, Foundations of Modern Probability, $2^nd$ edition, Springer, New York, 2002. doi: 10.1007/978-1-4757-4015-8.

[26]

S. KasjanG. Keller and M. Lemańczyk, Dynamics of $\mathcal B$-free sets: A view through the window, Int. Math. Res. Notices, 2019 (2019), 2690-2734.  doi: 10.1093/imrn/rnx196.

[27]

G. Keller, Tautness for sets of multiples and applications to $ {\mathcal B}$-free dynamics, Studia Mathematica, 247 (2019), 205-216.  doi: 10.4064/sm180305-9-4.

[28]

G. Keller and C. Richard, Dynamics on the graph of the torus parametrisation, Ergod. Th. & Dynam. Sys., 38 (2018), 1048-1085.  doi: 10.1017/etds.2016.53.

[29]

G. Keller and C. Richard, Periods and factors of weak model sets, Israel J. Math., 229 (2019), 85-132.  doi: 10.1007/s11856-018-1788-8.

[30]

H. B. Keynes and J. B. Robertson, Eigenvalue theorems in topological transformation groups, Transactions of the American Mathematical Society, 139 (1969), 359-369.  doi: 10.1090/S0002-9947-1969-0237748-5.

[31]

A. Kharazishvili, Topics in Measure Theory and Real Analysis, Atlantis Press / World Scientific, Amsterdam - Paris, 2009. doi: 10.2991/978-94-91216-36-7.

[32]

J. Kułaga-PrzymusM. Lemańczyk and B. Weiss, On invariant measures for ${ {\mathcal B}}$-free systems, Proceedings of the London Mathematical Society, 110 (2015), 1435-1474.  doi: 10.1112/plms/pdv017.

[33]

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.

[34]

M. K. Mentzen, Automorphisms of subshifts defined by $ {\mathcal B}$-free sets of integers., Colloquium Mathematicum, 147 (2017), 87-94.  doi: 10.4064/cm6927-5-2016.

[35]

R. V. Moody, Uniform distribution in model sets, Canad. Math. Bull., 45 (2002), 123-130.  doi: 10.4153/CMB-2002-015-3.

[36]

J. R. Munkres, Topology, Prentice Hall, $2^nd$ edition, 2000.

[37]

R. Peckner, Uniqueness of the measure of maximal entropy for the squarefree flow, Israel J. Math., 210 (2015), 335-357.  doi: 10.1007/s11856-015-1255-8.

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