# 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 & Continuous Dynamical Systems - A, 2012, 32 (1) : 293-301. doi: 10.3934/dcds.2012.32.293
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##### References:
 [1] Alfonso Artigue. Robustly N-expansive surface diffeomorphisms. Discrete & Continuous Dynamical Systems - A, 2016, 36 (5) : 2367-2376. doi: 10.3934/dcds.2016.36.2367 [2] Alfonso Artigue. Expansive flows of surfaces. Discrete & Continuous Dynamical Systems - A, 2013, 33 (2) : 505-525. doi: 10.3934/dcds.2013.33.505 [3] Jorge Groisman. Expansive homeomorphisms of the plane. Discrete & Continuous Dynamical Systems - A, 2011, 29 (1) : 213-239. doi: 10.3934/dcds.2011.29.213 [4] Alfonso Artigue. Lipschitz perturbations of expansive systems. Discrete & Continuous Dynamical Systems - A, 2015, 35 (5) : 1829-1841. doi: 10.3934/dcds.2015.35.1829 [5] Byung-Soo Lee. A convergence theorem of common fixed points of a countably infinite family of asymptotically quasi-$f_i$-expansive mappings in convex metric spaces. Numerical Algebra, Control & Optimization, 2013, 3 (3) : 557-565. doi: 10.3934/naco.2013.3.557 [6] Alfonso Artigue. Singular cw-expansive flows. Discrete & Continuous Dynamical Systems - A, 2017, 37 (6) : 2945-2956. doi: 10.3934/dcds.2017126 [7] Haritha C, Nikita Agarwal. Product of expansive Markov maps with hole. Discrete & Continuous Dynamical Systems - A, 2019, 39 (10) : 5743-5774. doi: 10.3934/dcds.2019252 [8] Jorge Groisman. Expansive and fixed point free homeomorphisms of the plane. Discrete & Continuous Dynamical Systems - A, 2012, 32 (5) : 1709-1721. doi: 10.3934/dcds.2012.32.1709 [9] Alfonso Artigue. Anomalous cw-expansive surface homeomorphisms. Discrete & Continuous Dynamical Systems - A, 2016, 36 (7) : 3511-3518. doi: 10.3934/dcds.2016.36.3511 [10] Martín Sambarino, José L. Vieitez. On $C^1$-persistently expansive homoclinic classes. Discrete & Continuous Dynamical Systems - A, 2006, 14 (3) : 465-481. doi: 10.3934/dcds.2006.14.465 [11] Martín Sambarino, José L. Vieitez. Robustly expansive homoclinic classes are generically hyperbolic. Discrete & Continuous Dynamical Systems - A, 2009, 24 (4) : 1325-1333. doi: 10.3934/dcds.2009.24.1325 [12] Keonhee Lee, Manseob Lee. Hyperbolicity of $C^1$-stably expansive homoclinic classes. Discrete & Continuous Dynamical Systems - A, 2010, 27 (3) : 1133-1145. doi: 10.3934/dcds.2010.27.1133 [13] Woochul Jung, Ngocthach Nguyen, Yinong Yang. Spectral decomposition for rescaling expansive flows with rescaled shadowing. Discrete & Continuous Dynamical Systems - A, 2020, 40 (4) : 2267-2283. doi: 10.3934/dcds.2020113 [14] Vianney Perchet, Marc Quincampoix. A differential game on Wasserstein space. Application to weak approachability with partial monitoring. Journal of Dynamics & Games, 2019, 6 (1) : 65-85. doi: 10.3934/jdg.2019005 [15] Jiguang Bao, Nguyen Lam, Guozhen Lu. Polyharmonic equations with critical exponential growth in the whole space $\mathbb{R}^{n}$. Discrete & Continuous Dynamical Systems - A, 2016, 36 (2) : 577-600. doi: 10.3934/dcds.2016.36.577 [16] Alfonso Artigue. Discrete and continuous topological dynamics: Fields of cross sections and expansive flows. Discrete & Continuous Dynamical Systems - A, 2016, 36 (11) : 5911-5927. doi: 10.3934/dcds.2016059 [17] Artur O. Lopes, Vladimir A. Rosas, Rafael O. Ruggiero. Cohomology and subcohomology problems for expansive, non Anosov geodesic flows. Discrete & Continuous Dynamical Systems - A, 2007, 17 (2) : 403-422. doi: 10.3934/dcds.2007.17.403 [18] Tatsuya Arai. The structure of dendrites constructed by pointwise P-expansive maps on the unit interval. Discrete & Continuous Dynamical Systems - A, 2016, 36 (1) : 43-61. doi: 10.3934/dcds.2016.36.43 [19] Lidong Wang, Hui Wang, Guifeng Huang. Minimal sets and $\omega$-chaos in expansive systems with weak specification property. Discrete & Continuous Dynamical Systems - A, 2015, 35 (3) : 1231-1238. doi: 10.3934/dcds.2015.35.1231 [20] Yong Xia, Ruey-Lin Sheu, Shu-Cherng Fang, Wenxun Xing. Double well potential function and its optimization in the $N$ -dimensional real space-part Ⅱ. Journal of Industrial & Management Optimization, 2017, 13 (3) : 1307-1328. doi: 10.3934/jimo.2016074

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