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New cw-expansive homeomorphisms of surfaces
Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 29 (2020) no. 2, pp. 221-246.

Dans ce travail on caractérise les extensions monotones d’homéomorphismes cw-expansifs d’espaces métriques compacts. Pour faire cela, on introduit la notion de “demi cw-expansivité” et on étudie son espace quotient naturel, notament dans le cas de surfaces compactes. On utilise ses résultats pour construire des exemples nouveaux d’homomémorphismes cw-expansifs avec un nombre infini de points fixes et dont l’ensemble errant est vide, dans le cadre des surfaces. Ces exemples sont des quotients de perturbations topologiques de pseudo-Anosov diffeomorphismes. Nous montrons également qu’il existe un homéomorphisme cw-expansif du 2-sphère avec la propriété shadowing.

In this article we characterize monotone extensions of cw-expansive homeomorphisms of compact metric spaces. For this purpose we introduce the notion “half cw-expansivity” and we study its natural quotient space, specially in the case of compact surfaces. These results are applied to construct new examples of cw-expansive homeomorphisms of compact surfaces with infinitely many fixed points and empty wandering set. These examples are quotients of topological perturbations of pseudo-Anosov diffeomorphisms. We also show that there is a cw-expansive homeomorphism with the shadowing property of the 2-sphere.

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DOI : https://doi.org/10.5802/afst.1630
@article{AFST_2020_6_29_2_221_0,
     author = {M. Achigar and A. Artigue and J. Vieitez},
     title = {New cw-expansive homeomorphisms of surfaces},
     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
     pages = {221--246},
     publisher = {Universit\'e Paul Sabatier, Toulouse},
     volume = {Ser. 6, 29},
     number = {2},
     year = {2020},
     doi = {10.5802/afst.1630},
     language = {en},
     url = {https://afst.centre-mersenne.org/articles/10.5802/afst.1630/}
}
M. Achigar; A. Artigue; J. Vieitez. New cw-expansive homeomorphisms of surfaces. Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 29 (2020) no. 2, pp. 221-246. doi : 10.5802/afst.1630. https://afst.centre-mersenne.org/articles/10.5802/afst.1630/

[1] Alfonso Artigue Hyper-expansive homeomorphisms, Publ. Mat. Urug., Volume 14 (2013), pp. 72-77 | MR 3235345 | Zbl 1309.37031

[2] Alfonso Artigue Anomalous cw-expansive surface homeomorphisms, Discrete Contin. Dyn. Syst., Volume 36 (2016) no. 7, pp. 3511-3518 | Article | MR 3485841 | Zbl 06560495

[3] Alfonso Artigue Robustly N-expansive surface diffeomorphisms, Discrete Contin. Dyn. Syst., Volume 36 (2016) no. 5, pp. 2367-2376 | Article | MR 3485401 | Zbl 1334.37025

[4] Alfonso Artigue Dendritations of surfaces, Ergodic Theory Dyn. Syst., Volume 38 (2018) no. 8, pp. 2860-2912 | Article | MR 3868017 | Zbl 1401.37016

[5] Alfonso Artigue; Dante Carrasco-Olivera A note on measure-expansive diffeomorphisms, J. Math. Anal. Appl., Volume 428 (2015) no. 1, pp. 713-716 | Article | MR 3327011 | Zbl 1376.37068

[6] Alfonso Artigue; María J. Pacifico; José L. Vieitez N-expansive homeomorphisms on surfaces, Commun. Contemp. Math., Volume 19 (2017) no. 1 | MR 3575918 | Zbl 1380.37089

[7] Rufus Bowen Entropy-expansive maps, Trans. Am. Math. Soc., Volume 164 (1972), pp. 323-331 | Article | MR 285689 | Zbl 0229.28011

[8] Marcelo Cerminara; Martin Sambarino Stable and unstable sets of C 0 perturbations of expansive homeomorphisms of surfaces, Nonlinearity, Volume 12 (1999) no. 2, pp. 321-332 | Article | MR 1677787 | Zbl 1031.37037

[9] Robert J. Daverman Decompositions of manifolds, Pure and Applied Mathematics, Volume 124, Academic Press Inc., 1986, xii+317 pages | MR 872468 | Zbl 0608.57002

[10] Lorenzo J. Díaz; Todd Fisher; María J. Pacifico; José L. Vieitez Entropy-expansiveness for partially hyperbolic diffeomorphisms, Discrete Contin. Dyn. Syst., Volume 32 (2012) no. 12, pp. 4195-4207 | MR 2966742 | Zbl 1256.37025

[11] Abbas Fakhari; Fatomeh H. Ghane Perturbation of continuum-wise expansive homeomorphisms, J. Dyn. Syst. Geom. Theor., Volume 3 (2005) no. 2, pp. 115-120 | MR 2180758 | Zbl 1082.54026

[12] Koichi Hiraide Expansive homeomorphisms of compact surfaces are pseudo-Anosov, Osaka J. Math., Volume 27 (1990) no. 1, pp. 117-162 | MR 1049828 | Zbl 0713.58042

[13] Hisao Kato Continuum-wise expansive homeomorphisms, Can. J. Math., Volume 45 (1993) no. 3, pp. 576-598 | Article | MR 1222517 | Zbl 0797.54047

[14] Kazimierz Kuratowski Topology. Vol. II, Academic Press Inc.; PWN-Polish Scientific Publishers, 1968, xiv+608 pages (new edition, revised and augmented, translated from the French by A. Kirkor) | Zbl 07227247

[15] Keonhee Lee; Seunghee Lee Continuum-wise expansive homeomorphisms with shadowing, J. Chungcheong Math. Soc., Volume 29 (2016) no. 1, pp. 151-155 | MR 3518042

[16] Jorge Lewowicz Persistence in expansive systems, Ergodic Theory Dyn. Syst., Volume 3 (1983) no. 4, pp. 567-578 | Article | MR 753924 | Zbl 0529.58021

[17] Jorge Lewowicz Expansive homeomorphisms of surfaces, Bol. Soc. Bras. Mat., Nova Sér., Volume 20 (1989) no. 1, pp. 113-133 | Article | MR 1129082 | Zbl 0753.58022

[18] Robert L. Moore Concerning upper semi-continuous collections of continua, Trans. Am. Math. Soc., Volume 27 (1925) no. 4, pp. 416-428 | Article | MR 1501320 | Zbl 51.0464.03

[19] Carlos A. Morales A generalization of expansivity, Discrete Contin. Dyn. Syst., Volume 32 (2012) no. 1, pp. 293-301 | Article | MR 2837062 | Zbl 1263.37017

[20] Carlos A. Morales; Víctor F. Sirvent Expansive measures, Publicações Matemáticas do IMPA, Instituto Nacional de Matemática Pura e Aplicada (IMPA), 2013, viii+89 pages (29 o Colóquio Brasileiro de Matemática. [29th Brazilian Mathematics Colloquium]) | Zbl 1283.37002

[21] Sam B. jun. Nadler Continuum Theory: An Introduction, Pure and Applied Mathematics, Marcel Dekker, Volume 158, Marcel Dekker, 1992 | Zbl 0757.54009

[22] John C. Oxtoby; Stanisław M. Ulam Measure-preserving homeomorphisms and metrical transitivity, Ann. Math., Volume 42 (1941), pp. 874-920 | Article | MR 5803 | Zbl 0063.06074

[23] María J. Pacifico; José L. Vieitez Entropy-expansiveness and domination for surface diffeomorphisms, Rev. Mat. Complut., Volume 21 (2008) no. 2, pp. 293-317 | MR 2441955 | Zbl 1151.37032

[24] María J. Pacifico; José L. Vieitez On measure expansive diffeomorphisms, Proc. Am. Math. Soc., Volume 143 (2015) no. 2, pp. 811-819 | Article | MR 3283667 | Zbl 1346.37022

[25] John H. Roberts; Norman E. Steenrod Monotone transformations of two-dimensional manifolds, Ann. Math., Princeton, Volume 39 (1938) no. 4, pp. 851-862 | Article | MR 1503441 | Zbl 64.1280.02

[26] Peter Walters An introduction to ergodic theory, Graduate Texts in Mathematics, Volume 79, Springer, 1982, ix+250 pages | MR 648108 | Zbl 0475.28009

[27] Raymond Louis Wilder Topology of Manifolds, Colloquium Publications, Volume 32, American Mathematical Society, 1949, ix+402 pages | MR 29491 | Zbl 0039.39602