# FINITE SETS WITH FAKE OBSERVABLE CARDINALITY

• Artigue, Alfonso
• Published : 2015.01.31
• 33 5

#### Abstract

Let X be a compact metric space and let |A| denote the cardinality of a set A. We prove that if $f:X{\rightarrow}X$ is a homeomorphism and ${\mid}X{\mid}={\infty}$, then for all ${\delta}$ > 0 there is $A{\subset}X$ such that |A| = 4 and for all $k{\in}\mathbb{Z}$ there are $x,y{\in}f^k(A)$, $x{\neq}y$, such that dist(x, y) < ${\delta}$. An observer that can only distinguish two points if their distance is grater than ${\delta}$, for sure will say that A has at most 3 points even knowing every iterate of A and that f is a homeomorphism. We show that for hyperexpansive homeomorphisms the same ${\delta}$-observer will not fail about the cardinality of A if we start with |A| = 3 instead of 4. Generalizations of this problem are considered via what we call (m, n)-expansiveness.

#### Keywords

topological dynamics;expansive homeomorphisms

#### References

1. A. Artigue, Hyper-expansive homeomorphisms, Publ. Mat. Urug. 14 (2013), 62-66.
2. A. Artigue, M. J. Pacifico, and J. L. Vieitez, N-expansive homeomorphisms on surfaces, Preprint, 2013.
3. R. Bowen, Entropy-expansive maps, Trans. Amer. Math. Soc. 164 (1972), 323-331. https://doi.org/10.1090/S0002-9947-1972-0285689-X
4. E. M. Coven and M. Keane, Every compact metric space that supports a positively expansive homeomorphism is finite, IMS Lecture Notes Monogr. Ser., Dynamics & Stochastics 4 (2006), 304-305.
5. J. R. Hertz, There are no stable points for continuum-wise expansive homeomorphisms, Pre. Mat. Urug. 65 (2002).
6. H. Kato, Continuum-wise expansive homeomorphisms, Canad. J. Math. 45 (1993), no. 3, 576-598. https://doi.org/10.4153/CJM-1993-030-4
7. J. Lewowicz, Dinamica de los homeomorfismos expansivos, Monografias del IMCA, 2003.
8. C. A. Morales, Measure expansive systems, Preprint IMPA, 2011.
9. C. A. Morales, A generalization of expansivity, Discrete Contin. Dyn. Syst. 32 (2012), no. 1, 293-301.
10. C. A. Morales and V. F. Sirvent, Expansive Measures, IMPA, 29o Coloq. Bras. Mat., 2013.
11. S. Nadler Jr., Hyperspaces of Sets, Marcel Dekker Inc. New York and Basel, 1978.
12. M. J. Pacifico and J. L. Vieitez, Entropy expansiveness and domination for surface diffeomorphisms, Rev. Mat. Complut. 21 (2008), no. 2, 293-317.
13. W. L. Reddy, Pointwise expansion homeomorphisms, J. Lond. Math. Soc. 2 (1970), 232-236.
14. K. Sakai, Continuum-wise expansive diffeomorphisms, Publ. Mat. 41 (1997), no. 2, 375-382. https://doi.org/10.5565/PUBLMAT_41297_04
15. S. Schwartzman, On transformation groups, Dissertation, Yale University, 1952.
16. W. R. Utz, Unstable homeomorphisms, Proc. Amer. Math. Soc. 1 (1950), no. 6, 769-774. https://doi.org/10.1090/S0002-9939-1950-0038022-3

#### Cited by

1. Levels of Generalized Expansiveness vol.29, pp.3, 2017, https://doi.org/10.1007/s10884-015-9502-6