# American Institute of Mathematical Sciences

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 and Continuous Dynamical Systems, 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), 71-114. doi: 10.1090/S0002-9947-1988-0927684-2. [2] N. Bourbaki, General Topology. Chapters 1-4, Translated from the French, reprint of the 1989 English translation, Elements of Mathematics (Berlin), Springer-Verlag, Berlin, 1998. [3] T. Downarowicz, Faces of simplexes of invariant measures, Israel J. Math., 165 (2008), 189-210. doi: 10.1007/s11856-008-1009-y. [4] T. Downarowicz, Entropy in dynamical systems, New Mathematical Monographs, 18, Cambridge University Press, Cambridge, 2011. doi: 10.1017/CBO9780511976155. [5] T. Downarowicz and J. Serafin, Possible entropy functions, Israel J. Math., 135 (2003), 221-250. doi: 10.1007/BF02776059. [6] S. Ferenczi, Systems of finite rank, Colloq. Math., 73 (1997), 35-65. [7] J. King, Joining-rank and the structure of finite rank mixing transformations, J. Analyse Math., 51 (1988), 182-227. doi: 10.1007/BF02791123. [8] I. Kornfeld and N. Ormes, Topological realizations of families of ergodic automorphisms, multitowers and orbit equivalence, Israel J. Math., 155 (2006), 335-357. doi: 10.1007/BF02773959. [9] D. S. Ornstein, D. J. Rudolph and B. Weiss, Equivalence of measure preserving transformations, Mem. Amer. Math. Soc., 37 (1982), xii+116 pp. doi: 10.1090/memo/0262. [10] H. L. Royden, Real Analysis, Third edition, Macmillan Publishing Company, New York, 1988. [11] W. H. Young, On a new method in the theory of integration, Proc. London Math. Soc., S2-9 (1911), 15-50. doi: 10.1112/plms/s2-9.1.15.

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##### References:
 [1] M. Boyle, D. Lind and D. Rudolph, The automorphism group of a shift of finite type, Trans. Amer. Math. Soc., 306 (1988), 71-114. doi: 10.1090/S0002-9947-1988-0927684-2. [2] N. Bourbaki, General Topology. Chapters 1-4, Translated from the French, reprint of the 1989 English translation, Elements of Mathematics (Berlin), Springer-Verlag, Berlin, 1998. [3] T. Downarowicz, Faces of simplexes of invariant measures, Israel J. Math., 165 (2008), 189-210. doi: 10.1007/s11856-008-1009-y. [4] T. Downarowicz, Entropy in dynamical systems, New Mathematical Monographs, 18, Cambridge University Press, Cambridge, 2011. doi: 10.1017/CBO9780511976155. [5] T. Downarowicz and J. Serafin, Possible entropy functions, Israel J. Math., 135 (2003), 221-250. doi: 10.1007/BF02776059. [6] S. Ferenczi, Systems of finite rank, Colloq. Math., 73 (1997), 35-65. [7] J. King, Joining-rank and the structure of finite rank mixing transformations, J. Analyse Math., 51 (1988), 182-227. doi: 10.1007/BF02791123. [8] I. Kornfeld and N. Ormes, Topological realizations of families of ergodic automorphisms, multitowers and orbit equivalence, Israel J. Math., 155 (2006), 335-357. doi: 10.1007/BF02773959. [9] D. S. Ornstein, D. J. Rudolph and B. Weiss, Equivalence of measure preserving transformations, Mem. Amer. Math. Soc., 37 (1982), xii+116 pp. doi: 10.1090/memo/0262. [10] H. L. Royden, Real Analysis, Third edition, Macmillan Publishing Company, New York, 1988. [11] W. H. Young, On a new method in the theory of integration, Proc. London Math. Soc., S2-9 (1911), 15-50. doi: 10.1112/plms/s2-9.1.15.
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