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December  2021, 41(12): 5915-5942. doi: 10.3934/dcds.2021101

Maximal chain continuous factor

Noriaki Kawaguchi

Faculty of Science and Technology, Keio University, 3-14-1 Hiyoshi, Kohoku-ku, Yokohama, Kanagawa 223-8522, Japan

JSPS Research Fellow

Received  July 2020 Revised  May 2021 Published  December 2021 Early access  June 2021

For any continuous self-map of a compact metric space, we define, prove the existence, and give an explicit expression of a maximal chain continuous factor. For the purpose, we exploit a chain proximal relation and its extension. An example is given to illustrate a difference of the two relations. An alternative proof of a result on the odometers and the regular recurrence is given. Also, we provide an example of a calculation of the maximal chain continuous factor for generic homeomorphism of the Cantor set.

Keywords: Chain continuous, equicontinuous, totally disconnected, maximal, factor, odometer, regularly recurrent.
Mathematics Subject Classification: Primary: 37B02; Secondary: 37B05, 37B65.
Citation: Noriaki Kawaguchi. Maximal chain continuous factor. Discrete & Continuous Dynamical Systems, 2021, 41 (12) : 5915-5942. doi: 10.3934/dcds.2021101
References:
[1]

E. Akin, The General Topology of Dynamical Systems. Graduate Studies in Mathematics, 1. American Mathematical Society, Providence, RI, 1993. doi: 10.1090/gsm/001.  Google Scholar

[2]

E. Akin, On chain continuity, Discrete Contin. Dynam. Systems, 2 (1996), 111-120.  doi: 10.3934/dcds.1996.2.111.  Google Scholar

[3]

E. Akin and E. Glasner, Residual properties and almost equicontinuity, J. Anal. Math., 84 (2001), 243-286.  doi: 10.1007/BF02788112.  Google Scholar

[4]

E. Akin, E. Glasner and B. Weiss, Generically there is but one self homeomorphism of the Cantor set, Trans. Amer. Math. Soc., 360 (2008), 3613-3630.  doi: 10.1090/S0002-9947-08-04450-4.  Google Scholar

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E. Akin and J. Wiseman, Varieties of mixing, Trans. Amer. Math. Soc., 372 (2019), 4359-4390.  doi: 10.1090/tran/7681.  Google Scholar

[6]

N. Aoki and K. Hiraide, Topological Theory of Dynamical Systems. Recent Advances, North-Holland Mathematical Library, 52. North-Holland Publishing Co., Amsterdam, 1994.  Google Scholar

[7]

J. Auslander, Minimal Flows and their Extensions, North-Holland Mathematics Studies, 153. North-Holland Publishing Co., Amsterdam, 1988.  Google Scholar

[8]

J. Auslander, Two folk theorems in topological dynamics, Eur. J. Math., 2 (2016), 539-543.  doi: 10.1007/s40879-016-0097-1.  Google Scholar

[9]

N. C. Bernardes Jr. and U. B. Darji, Graph theoretic structure of maps of the Cantor space, Adv. Math., 231 (2012), 1655-1680.  doi: 10.1016/j.aim.2012.05.024.  Google Scholar

[10]

L. Blokh and J. Keesling, A characterization of adding machine maps, Topology Appl., 140 (2004), 151-161.  doi: 10.1016/j.topol.2003.07.006.  Google Scholar

[11]

J. Buescu, M. Kulczycki and I. Stewart, Liapunov stability and adding machines revisited, Dyn. Syst., 21 (2006), 379-384.  doi: 10.1080/14689360600649815.  Google Scholar

[12]

J. Buescu and I. Stewart, Liapunov stability and adding machines, Ergodic Theory Dynam. Systems, 15 (1995), 271-290.  doi: 10.1017/S0143385700008373.  Google Scholar

[13]

T. Downarowicz, Survey of odometers and Toeplitz flows, Algebraic and Topological Dynamics, 7–37, Contemp. Math., 385, Amer. Math. Soc., Providence, RI, 2005. doi: 10.1090/conm/385/07188.  Google Scholar

[14]

M. W. Hirsch and M. Hurley, Connected components of attractors and other stable sets, Aequationes Math., 53 (1997), 308-323.  doi: 10.1007/BF02215978.  Google Scholar

[15]

M. Hochman, Genericity in topological dynamics, Ergodic Theory Dynam. Systems, 28 (2008), 125-165.  doi: 10.1017/S0143385707000521.  Google Scholar

[16]

W. Huang and X. Ye, Devaney's chaos or 2-scattering implies Li-Yorke's chaos, Topology Appl., 117 (2002), 259-272.  doi: 10.1016/S0166-8641(01)00025-6.  Google Scholar

[17]

N. Kawaguchi, Quantitative shadowable points, Dyn. Syst., 32 (2017), 504-518.  doi: 10.1080/14689367.2017.1280664.  Google Scholar

[18]

N. Kawaguchi, Properties of shadowable points: chaos and equicontinuity, Bull. Braz. Math. Soc. (N.S.), 48 (2017), 599-622.  doi: 10.1007/s00574-017-0033-0.  Google Scholar

[19]

N. Kawaguchi, Distributionally chaotic maps are $C^0$-dense, Proc. Amer. Math. Soc., 147 (2019), 5339-5348.  doi: 10.1090/proc/14696.  Google Scholar

[20]

A.S. Kechris and C. Rosendal, Turbulence, amalgamation, and generic automorphisms of homogeneous structures, Proc. Lond. Math. Soc., 94 (2007), 302-350.  doi: 10.1112/plms/pdl007.  Google Scholar

[21]

J. Kupka and P. Oprocha, On the dynamics of generic maps on the Cantor set, Topology Appl., 263 (2019), 330-342.  doi: 10.1016/j.topol.2019.05.029.  Google Scholar

[22]

P. Kůrka, Topological and Symbolic Dynamics, Societe Mathematique de France, Paris, 2003.  Google Scholar

[23]

J. H. Mai, The structure of equicontinuous maps, Trans. Amer. Math. Soc., 355 (2003), 4125-4136.  doi: 10.1090/S0002-9947-03-03339-7.  Google Scholar

[24]

D. Richeson and J. Wiseman, Chain recurrence rates and topological entropy, Topology Appl., 156 (2008), 251-261.  doi: 10.1016/j.topol.2008.07.005.  Google Scholar

show all references

References:
[1]

E. Akin, The General Topology of Dynamical Systems. Graduate Studies in Mathematics, 1. American Mathematical Society, Providence, RI, 1993. doi: 10.1090/gsm/001.  Google Scholar

[2]

E. Akin, On chain continuity, Discrete Contin. Dynam. Systems, 2 (1996), 111-120.  doi: 10.3934/dcds.1996.2.111.  Google Scholar

[3]

E. Akin and E. Glasner, Residual properties and almost equicontinuity, J. Anal. Math., 84 (2001), 243-286.  doi: 10.1007/BF02788112.  Google Scholar

[4]

E. Akin, E. Glasner and B. Weiss, Generically there is but one self homeomorphism of the Cantor set, Trans. Amer. Math. Soc., 360 (2008), 3613-3630.  doi: 10.1090/S0002-9947-08-04450-4.  Google Scholar

[5]

E. Akin and J. Wiseman, Varieties of mixing, Trans. Amer. Math. Soc., 372 (2019), 4359-4390.  doi: 10.1090/tran/7681.  Google Scholar

[6]

N. Aoki and K. Hiraide, Topological Theory of Dynamical Systems. Recent Advances, North-Holland Mathematical Library, 52. North-Holland Publishing Co., Amsterdam, 1994.  Google Scholar

[7]

J. Auslander, Minimal Flows and their Extensions, North-Holland Mathematics Studies, 153. North-Holland Publishing Co., Amsterdam, 1988.  Google Scholar

[8]

J. Auslander, Two folk theorems in topological dynamics, Eur. J. Math., 2 (2016), 539-543.  doi: 10.1007/s40879-016-0097-1.  Google Scholar

[9]

N. C. Bernardes Jr. and U. B. Darji, Graph theoretic structure of maps of the Cantor space, Adv. Math., 231 (2012), 1655-1680.  doi: 10.1016/j.aim.2012.05.024.  Google Scholar

[10]

L. Blokh and J. Keesling, A characterization of adding machine maps, Topology Appl., 140 (2004), 151-161.  doi: 10.1016/j.topol.2003.07.006.  Google Scholar

[11]

J. Buescu, M. Kulczycki and I. Stewart, Liapunov stability and adding machines revisited, Dyn. Syst., 21 (2006), 379-384.  doi: 10.1080/14689360600649815.  Google Scholar

[12]

J. Buescu and I. Stewart, Liapunov stability and adding machines, Ergodic Theory Dynam. Systems, 15 (1995), 271-290.  doi: 10.1017/S0143385700008373.  Google Scholar

[13]

T. Downarowicz, Survey of odometers and Toeplitz flows, Algebraic and Topological Dynamics, 7–37, Contemp. Math., 385, Amer. Math. Soc., Providence, RI, 2005. doi: 10.1090/conm/385/07188.  Google Scholar

[14]

M. W. Hirsch and M. Hurley, Connected components of attractors and other stable sets, Aequationes Math., 53 (1997), 308-323.  doi: 10.1007/BF02215978.  Google Scholar

[15]

M. Hochman, Genericity in topological dynamics, Ergodic Theory Dynam. Systems, 28 (2008), 125-165.  doi: 10.1017/S0143385707000521.  Google Scholar

[16]

W. Huang and X. Ye, Devaney's chaos or 2-scattering implies Li-Yorke's chaos, Topology Appl., 117 (2002), 259-272.  doi: 10.1016/S0166-8641(01)00025-6.  Google Scholar

[17]

N. Kawaguchi, Quantitative shadowable points, Dyn. Syst., 32 (2017), 504-518.  doi: 10.1080/14689367.2017.1280664.  Google Scholar

[18]

N. Kawaguchi, Properties of shadowable points: chaos and equicontinuity, Bull. Braz. Math. Soc. (N.S.), 48 (2017), 599-622.  doi: 10.1007/s00574-017-0033-0.  Google Scholar

[19]

N. Kawaguchi, Distributionally chaotic maps are $C^0$-dense, Proc. Amer. Math. Soc., 147 (2019), 5339-5348.  doi: 10.1090/proc/14696.  Google Scholar

[20]

A.S. Kechris and C. Rosendal, Turbulence, amalgamation, and generic automorphisms of homogeneous structures, Proc. Lond. Math. Soc., 94 (2007), 302-350.  doi: 10.1112/plms/pdl007.  Google Scholar

[21]

J. Kupka and P. Oprocha, On the dynamics of generic maps on the Cantor set, Topology Appl., 263 (2019), 330-342.  doi: 10.1016/j.topol.2019.05.029.  Google Scholar

[22]

P. Kůrka, Topological and Symbolic Dynamics, Societe Mathematique de France, Paris, 2003.  Google Scholar

[23]

J. H. Mai, The structure of equicontinuous maps, Trans. Amer. Math. Soc., 355 (2003), 4125-4136.  doi: 10.1090/S0002-9947-03-03339-7.  Google Scholar

[24]

D. Richeson and J. Wiseman, Chain recurrence rates and topological entropy, Topology Appl., 156 (2008), 251-261.  doi: 10.1016/j.topol.2008.07.005.  Google Scholar

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