# American Institute of Mathematical Sciences

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January  2012, 32(1): 293-301. doi: 10.3934/dcds.2012.32.293

## A generalization of expansivity

 1 Instituto de Matemàtica, Universidade Federal do Rio de Janeiro, C. P. 68530, CEP 21945-970, Rio de Janeiro, Brazil

Received  July 2010 Revised  January 2011 Published  September 2011

We study dynamical systems for which at most $n$ orbits can accompany a given arbitrary orbit. For simplicity we call them $n$-expansive (or positively $n$-expansive if positive orbits are considered instead). We prove that these systems can satisfy properties of expansive systems or not. For instance, unlike positively expansive maps [3], positively $n$-expansive homeomorphisms may exist on certain infinite compact metric spaces. We also prove that a map (resp. bijective map) is positively $n$-expansive (resp. $n$-expansive) if and only if it is so outside finitely many points. Finally, we prove that a homeomorphism on a compact metric space is $n$-expansive if and only if it is so outside finitely many orbits. These last resuls extends previous ones for expansive systems [2],[11],[12].
Citation: Carlos Arnoldo Morales. A generalization of expansivity. Discrete and Continuous Dynamical Systems, 2012, 32 (1) : 293-301. doi: 10.3934/dcds.2012.32.293
##### References:
 [1] R. Bowen, Entropy-expansive maps, Trans. Amer. Math. Soc., 164 (1972), 323-331. doi: 10.1090/S0002-9947-1972-0285689-X. [2] B. F. Bryant, Expansive self-homeomorphisms of a compact metric space, Amer. Math. Monthly, 69 (1962), 386-391. doi: 10.2307/2312129. [3] E. M. Coven and M. Keane, Every compact metric space that supports a positively expansive homeomorphism is finite, Dynamics & Stochastics, IMS Lecture Notes Monogr. Ser., 48, Inst. Math. Statist., Beachwood, OH, 2006, 304-305. [4] J. Dydak and C. S. Hoffland, An alternative definition of coarse structures, Topology Appl., 155 (2008), 1013-1021. doi: 10.1016/j.topol.2008.01.002. [5] M. Eisenberg, Expansive transformation semigroups of endomorphisms, Fund. Math., 59 (1966), 313-321. [6] A. Katok and B. Hasselblatt, "Introduction to the Modern Theory of Dynamical Systems," with a supplementary chapter by Katok and Leonardo Mendoza, Encyclopedia of Mathematics and its Applications, 54, Cambridge University Press, Cambridge, 1995. [7] H. Kato, Continuum-wise expansive homeomorphisms, Canad. J. Math., 45 (1993), 576-598. doi: 10.4153/CJM-1993-030-4. [8] C. Morales, Measure-expansive systems, Preprint (2010) submitted. [9] W. L. Reddy, Pointwise expansion homeomorphisms, J. London Math. Soc., 2 (1970), 232-236. doi: 10.1112/jlms/s2-2.2.232. [10] W. R. Utz, Unstable homeomorphisms, Proc. Amer. Math. Soc., 1 (1950), 769-774. doi: 10.1090/S0002-9939-1950-0038022-3. [11] W. R. Utz, Expansive mappings, Proceedings of the 1978 Topology Conference (Univ. Oklahoma, Norman, Okla.), I. Topology Proc., 3 (1978), 221-226. [12] R. K. Williams, On expansive homeomorphisms, Amer. Math. Monthly, 76 (1969), 176-178. doi: 10.2307/2317269.

show all references

##### References:
 [1] R. Bowen, Entropy-expansive maps, Trans. Amer. Math. Soc., 164 (1972), 323-331. doi: 10.1090/S0002-9947-1972-0285689-X. [2] B. F. Bryant, Expansive self-homeomorphisms of a compact metric space, Amer. Math. Monthly, 69 (1962), 386-391. doi: 10.2307/2312129. [3] E. M. Coven and M. Keane, Every compact metric space that supports a positively expansive homeomorphism is finite, Dynamics & Stochastics, IMS Lecture Notes Monogr. Ser., 48, Inst. Math. Statist., Beachwood, OH, 2006, 304-305. [4] J. Dydak and C. S. Hoffland, An alternative definition of coarse structures, Topology Appl., 155 (2008), 1013-1021. doi: 10.1016/j.topol.2008.01.002. [5] M. Eisenberg, Expansive transformation semigroups of endomorphisms, Fund. Math., 59 (1966), 313-321. [6] A. Katok and B. Hasselblatt, "Introduction to the Modern Theory of Dynamical Systems," with a supplementary chapter by Katok and Leonardo Mendoza, Encyclopedia of Mathematics and its Applications, 54, Cambridge University Press, Cambridge, 1995. [7] H. Kato, Continuum-wise expansive homeomorphisms, Canad. J. Math., 45 (1993), 576-598. doi: 10.4153/CJM-1993-030-4. [8] C. Morales, Measure-expansive systems, Preprint (2010) submitted. [9] W. L. Reddy, Pointwise expansion homeomorphisms, J. London Math. Soc., 2 (1970), 232-236. doi: 10.1112/jlms/s2-2.2.232. [10] W. R. Utz, Unstable homeomorphisms, Proc. Amer. Math. Soc., 1 (1950), 769-774. doi: 10.1090/S0002-9939-1950-0038022-3. [11] W. R. Utz, Expansive mappings, Proceedings of the 1978 Topology Conference (Univ. Oklahoma, Norman, Okla.), I. Topology Proc., 3 (1978), 221-226. [12] R. K. Williams, On expansive homeomorphisms, Amer. Math. Monthly, 76 (1969), 176-178. doi: 10.2307/2317269.
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