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On the number of invariant measures for random expanding maps in higher dimensions
Maximal chain continuous factor
Faculty of Science and Technology, Keio University, 3-14-1 Hiyoshi, Kohoku-ku, Yokohama, Kanagawa 223-8522, Japan |
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.
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. |
| [2] |
E. Akin,
On chain continuity, Discrete Contin. Dynam. Systems, 2 (1996), 111-120.
doi: 10.3934/dcds.1996.2.111. |
| [3] |
E. Akin and E. Glasner,
Residual properties and almost equicontinuity, J. Anal. Math., 84 (2001), 243-286.
doi: 10.1007/BF02788112. |
| [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. |
| [5] |
E. Akin and J. Wiseman,
Varieties of mixing, Trans. Amer. Math. Soc., 372 (2019), 4359-4390.
doi: 10.1090/tran/7681. |
| [6] |
N. Aoki and K. Hiraide, Topological Theory of Dynamical Systems. Recent Advances, North-Holland Mathematical Library, 52. North-Holland Publishing Co., Amsterdam, 1994. |
| [7] |
J. Auslander, Minimal Flows and their Extensions, North-Holland Mathematics Studies, 153. North-Holland Publishing Co., Amsterdam, 1988. |
| [8] |
J. Auslander,
Two folk theorems in topological dynamics, Eur. J. Math., 2 (2016), 539-543.
doi: 10.1007/s40879-016-0097-1. |
| [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. |
| [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. |
| [11] |
J. Buescu, M. Kulczycki and I. Stewart,
Liapunov stability and adding machines revisited, Dyn. Syst., 21 (2006), 379-384.
doi: 10.1080/14689360600649815. |
| [12] |
J. Buescu and I. Stewart,
Liapunov stability and adding machines, Ergodic Theory Dynam. Systems, 15 (1995), 271-290.
doi: 10.1017/S0143385700008373. |
| [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. |
| [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. |
| [15] |
M. Hochman,
Genericity in topological dynamics, Ergodic Theory Dynam. Systems, 28 (2008), 125-165.
doi: 10.1017/S0143385707000521. |
| [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. |
| [17] |
N. Kawaguchi,
Quantitative shadowable points, Dyn. Syst., 32 (2017), 504-518.
doi: 10.1080/14689367.2017.1280664. |
| [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. |
| [19] |
N. Kawaguchi,
Distributionally chaotic maps are $C^0$-dense, Proc. Amer. Math. Soc., 147 (2019), 5339-5348.
doi: 10.1090/proc/14696. |
| [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. |
| [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. |
| [22] |
P. Kůrka, Topological and Symbolic Dynamics, Societe Mathematique de France, Paris, 2003. |
| [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. |
| [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. |
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. |
| [2] |
E. Akin,
On chain continuity, Discrete Contin. Dynam. Systems, 2 (1996), 111-120.
doi: 10.3934/dcds.1996.2.111. |
| [3] |
E. Akin and E. Glasner,
Residual properties and almost equicontinuity, J. Anal. Math., 84 (2001), 243-286.
doi: 10.1007/BF02788112. |
| [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. |
| [5] |
E. Akin and J. Wiseman,
Varieties of mixing, Trans. Amer. Math. Soc., 372 (2019), 4359-4390.
doi: 10.1090/tran/7681. |
| [6] |
N. Aoki and K. Hiraide, Topological Theory of Dynamical Systems. Recent Advances, North-Holland Mathematical Library, 52. North-Holland Publishing Co., Amsterdam, 1994. |
| [7] |
J. Auslander, Minimal Flows and their Extensions, North-Holland Mathematics Studies, 153. North-Holland Publishing Co., Amsterdam, 1988. |
| [8] |
J. Auslander,
Two folk theorems in topological dynamics, Eur. J. Math., 2 (2016), 539-543.
doi: 10.1007/s40879-016-0097-1. |
| [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. |
| [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. |
| [11] |
J. Buescu, M. Kulczycki and I. Stewart,
Liapunov stability and adding machines revisited, Dyn. Syst., 21 (2006), 379-384.
doi: 10.1080/14689360600649815. |
| [12] |
J. Buescu and I. Stewart,
Liapunov stability and adding machines, Ergodic Theory Dynam. Systems, 15 (1995), 271-290.
doi: 10.1017/S0143385700008373. |
| [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. |
| [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. |
| [15] |
M. Hochman,
Genericity in topological dynamics, Ergodic Theory Dynam. Systems, 28 (2008), 125-165.
doi: 10.1017/S0143385707000521. |
| [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. |
| [17] |
N. Kawaguchi,
Quantitative shadowable points, Dyn. Syst., 32 (2017), 504-518.
doi: 10.1080/14689367.2017.1280664. |
| [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. |
| [19] |
N. Kawaguchi,
Distributionally chaotic maps are $C^0$-dense, Proc. Amer. Math. Soc., 147 (2019), 5339-5348.
doi: 10.1090/proc/14696. |
| [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. |
| [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. |
| [22] |
P. Kůrka, Topological and Symbolic Dynamics, Societe Mathematique de France, Paris, 2003. |
| [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. |
| [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. |
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