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  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 & Continuous Dynamical Systems - A, 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.  Google Scholar

[2]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

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

[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.  Google Scholar

[8]

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

[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.  Google Scholar

[10]

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

[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.  Google Scholar

[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.  Google Scholar

[13]

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

[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.  Google Scholar

[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.  Google Scholar

[16]

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

[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.  Google Scholar

[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.  Google Scholar

[19]

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

[20]

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

[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.  Google Scholar

[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.  Google Scholar

[23]

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

[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.  Google Scholar

[25]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[35]

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

[36]

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

[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.  Google Scholar

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.  Google Scholar

[2]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

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

[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.  Google Scholar

[8]

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

[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.  Google Scholar

[10]

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

[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.  Google Scholar

[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.  Google Scholar

[13]

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

[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.  Google Scholar

[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.  Google Scholar

[16]

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

[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.  Google Scholar

[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.  Google Scholar

[19]

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

[20]

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

[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.  Google Scholar

[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.  Google Scholar

[23]

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

[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.  Google Scholar

[25]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[35]

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

[36]

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

[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.  Google Scholar

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