July  2014, 34(7): 2741-2750. doi: 10.3934/dcds.2014.34.2741

Rank as a function of measure

1. 

Institute of Mathematics and Computer Science, Wroclaw University of Technology, Wybrzeże Wyspiańskiego 27, 50-370 Wroc law, Poland, Poland

2. 

Institute of Mathematics, Polish Academy of Sciences, ul. Śniadeckich 8, 00-956 Warszawa, Poland

Received  June 2012 Revised  October 2013 Published  December 2013

We establish certain topological properties of rank understood as a function on the set of invariant measures on a topological dynamical system. To be exact, we show that rank is of Young class LU (i.e., it is the limit of an increasing sequence of upper semicontinuous functions).
Citation: Tomasz Downarowicz, Yonatan Gutman, Dawid Huczek. Rank as a function of measure. Discrete & Continuous Dynamical Systems - A, 2014, 34 (7) : 2741-2750. doi: 10.3934/dcds.2014.34.2741
References:
[1]

M. Boyle, D. Lind and D. Rudolph, The automorphism group of a shift of finite type,, Trans. Amer. Math. Soc., 306 (1988), 71. doi: 10.1090/S0002-9947-1988-0927684-2. Google Scholar

[2]

N. Bourbaki, General Topology. Chapters 1-4,, Translated from the French, (1989). Google Scholar

[3]

T. Downarowicz, Faces of simplexes of invariant measures,, Israel J. Math., 165 (2008), 189. doi: 10.1007/s11856-008-1009-y. Google Scholar

[4]

T. Downarowicz, Entropy in dynamical systems,, New Mathematical Monographs, (2011). doi: 10.1017/CBO9780511976155. Google Scholar

[5]

T. Downarowicz and J. Serafin, Possible entropy functions,, Israel J. Math., 135 (2003), 221. doi: 10.1007/BF02776059. Google Scholar

[6]

S. Ferenczi, Systems of finite rank,, Colloq. Math., 73 (1997), 35. Google Scholar

[7]

J. King, Joining-rank and the structure of finite rank mixing transformations,, J. Analyse Math., 51 (1988), 182. doi: 10.1007/BF02791123. Google Scholar

[8]

I. Kornfeld and N. Ormes, Topological realizations of families of ergodic automorphisms, multitowers and orbit equivalence,, Israel J. Math., 155 (2006), 335. doi: 10.1007/BF02773959. Google Scholar

[9]

D. S. Ornstein, D. J. Rudolph and B. Weiss, Equivalence of measure preserving transformations,, Mem. Amer. Math. Soc., 37 (1982). doi: 10.1090/memo/0262. Google Scholar

[10]

H. L. Royden, Real Analysis,, Third edition, (1988). Google Scholar

[11]

W. H. Young, On a new method in the theory of integration,, Proc. London Math. Soc., S2-9 (1911), 2. doi: 10.1112/plms/s2-9.1.15. Google Scholar

show all references

References:
[1]

M. Boyle, D. Lind and D. Rudolph, The automorphism group of a shift of finite type,, Trans. Amer. Math. Soc., 306 (1988), 71. doi: 10.1090/S0002-9947-1988-0927684-2. Google Scholar

[2]

N. Bourbaki, General Topology. Chapters 1-4,, Translated from the French, (1989). Google Scholar

[3]

T. Downarowicz, Faces of simplexes of invariant measures,, Israel J. Math., 165 (2008), 189. doi: 10.1007/s11856-008-1009-y. Google Scholar

[4]

T. Downarowicz, Entropy in dynamical systems,, New Mathematical Monographs, (2011). doi: 10.1017/CBO9780511976155. Google Scholar

[5]

T. Downarowicz and J. Serafin, Possible entropy functions,, Israel J. Math., 135 (2003), 221. doi: 10.1007/BF02776059. Google Scholar

[6]

S. Ferenczi, Systems of finite rank,, Colloq. Math., 73 (1997), 35. Google Scholar

[7]

J. King, Joining-rank and the structure of finite rank mixing transformations,, J. Analyse Math., 51 (1988), 182. doi: 10.1007/BF02791123. Google Scholar

[8]

I. Kornfeld and N. Ormes, Topological realizations of families of ergodic automorphisms, multitowers and orbit equivalence,, Israel J. Math., 155 (2006), 335. doi: 10.1007/BF02773959. Google Scholar

[9]

D. S. Ornstein, D. J. Rudolph and B. Weiss, Equivalence of measure preserving transformations,, Mem. Amer. Math. Soc., 37 (1982). doi: 10.1090/memo/0262. Google Scholar

[10]

H. L. Royden, Real Analysis,, Third edition, (1988). Google Scholar

[11]

W. H. Young, On a new method in the theory of integration,, Proc. London Math. Soc., S2-9 (1911), 2. doi: 10.1112/plms/s2-9.1.15. Google Scholar

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