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Shadowing is generic---a continuous map case
1. | Jagiellonian University, Faculty of Mathematics and Computer Science, Institute of Mathematics, Department of Applied Mathematics, ul. Łojasiewicza 6, 30-348 Kraków, Poland, Poland |
2. | Faculty of Applied Mathematics, AGH University of Science and Technology, al. Mickiewicza 30, 30-059 Kraków |
3. | Centre of Mathematics, University of Minho, Campus de Gualtar, 4710-057 Braga, Portugal |
References:
[1] |
F. Abdenur and L. Díaz, Pseudo-orbit shadowing in the $C^1$ topology, Discrete Contin. Dyn. Syst., 17 (2007), 223-245. |
[2] |
D. V. Anosov, Geodesic Flows on Closed Riemann Manifolds with Negative Curvature, (translated from the Russian by S. Feder), American Mathematical Society, Providence, R.I., 1969. |
[3] |
N. Aoki and K. Hiraide, Topological Theory of Dynamical Systems. Recent Advances, North-Holland Publishing Co., Amsterdam, 1994. |
[4] |
R. H. Bing, An alternative proof that $3$-manifolds can be triangulated, Ann. of Math., 69 (1959), 37-65.
doi: 10.2307/1970092. |
[5] |
C. Bonatti, L. J. Díaz and G. Turcat, Pas de "shadowing lemma'' pour des dynamiques partiellement hyperboliques (French) [There is no shadowing lemma for partially hyperbolic dynamics], C. R. Acad. Sci. Paris Sér. I Math., 330 (2000), 587-592.
doi: 10.1016/S0764-4442(00)00215-9. |
[6] |
R. Bowen, $\omega$-limit sets for Axiom A diffeomorphisms, J. Differential Equations, 18 (1975), 333-339.
doi: 10.1016/0022-0396(75)90065-0. |
[7] |
S. S. Cairns, Triangulation of the manifold of class one, Bull. Amer. Math. Soc., 41 (1935), 549-552.
doi: 10.1090/S0002-9904-1935-06140-3. |
[8] |
L. Chen, Linking and the shadowing property for piecewise monotone maps, Proc. Amer. Math. Soc., 113 (1991), 251-263.
doi: 10.1090/S0002-9939-1991-1079695-2. |
[9] |
B. A. Coomes, H. Koçak and K. J. Palmer, Periodic shadowing, in Chaotic numerics(eds. P.E. Kloeden and K.J. Palmer), Amer. Math. Soc., Providence, R.I., (1994), 115-130.
doi: 10.1090/conm/172/01801. |
[10] |
R. M. Corless and S. Yu. Pilyugin, Approximate and real trajectories for generic dynamical systems, J. Math. Anal. Appl., 189 (1995), 409-423.
doi: 10.1006/jmaa.1995.1027. |
[11] |
E. M. Coven, I. Kan and J. A. Yorke, Pseudo-orbit shadowing in the family of tent maps, Trans. Amer. Math. Soc., 308 (1988), 227-241.
doi: 10.1090/S0002-9947-1988-0946440-2. |
[12] |
S. Crovisier, Periodic orbits and chain-transitive sets of $C^1$-diffeomorphisms, Publ. Math. Inst. Hautes Études Sci., 104 (2006), 87-141.
doi: 10.1007/s10240-006-0002-4. |
[13] |
J. Dugundji, An extension of Tietze's theorem, Pacific J. Math., 1 (1951), 353-367.
doi: 10.2140/pjm.1951.1.353. |
[14] |
J. Franks and C. A. Robinson, A quasi-Anosov diffeomorphism that is not Anosov, Trans. Amer. Math. Soc., 223 (1976), 267-278.
doi: 10.1090/S0002-9947-1976-0423420-9. |
[15] |
M. H. Freedman, The topology of four-dimensional manifolds, J. Differential Geom., 17 (1982), 357-453. |
[16] |
R. C. Kirby and L. C. Siebenmann, On the triangulation of manifolds and the Hauptvermutung, Bull. Amer. Math. Soc., 75 (1969), 742-749.
doi: 10.1090/S0002-9904-1969-12271-8. |
[17] |
R. C. Kirby and L. C. Siebenmann, Foundational Essays on Topological Manifolds, Smoothings, and Triangulations, Princeton University Press, Princeton, N.J., 1977. |
[18] |
U. Kirchgraber, U. Manz and D. Stoffer, Rigorous proof of chaotic behaviour in a dumbbell satellite model, J. Math. Anal. Appl., 251 (2000), 897-911.
doi: 10.1006/jmaa.2000.7143. |
[19] |
U. Kirchgraber and D. Stoffer, Possible chaotic motion of comets in the Sun-Jupiter system-a computer-assisted approach based on shadowing, Nonlinearity, 17 (2004), 281-300.
doi: 10.1088/0951-7715/17/1/016. |
[20] |
P. E. Kloeden and J. Ombach, Hyperbolic homeomorphisms are bishadowing, Ann. Polon. Math., 65 (1997), 171-177. |
[21] |
P. Kościelniak, On genericity of shadowing and periodic shadowing property, J. Math. Anal. Appl., 310 (2005), 188-196.
doi: 10.1016/j.jmaa.2005.01.053. |
[22] |
P. Kościelniak, Generic properties of $\mathbb Z^{2}$-actions on the interval, Topology Appl., 154 (2007), 2672-2677.
doi: 10.1016/j.topol.2007.05.001. |
[23] |
P. Kościelniak and M. Mazur, On $C^0$ genericity of various shadowing properties, Discrete Contin. Dyn. Syst., 12 (2005), 523-530. |
[24] |
P. Kościelniak and M. Mazur, Chaos and the shadowing property, Topology Appl., 154 (2007), 2553-2557.
doi: 10.1016/j.topol.2006.06.010. |
[25] |
P. Kościelniak and M. Mazur, Genericity of inverse shadowing property, J. Difference Equ. Appl., 16 (2010), 667-674.
doi: 10.1080/10236190903213464. |
[26] |
J. Llibre and M. Misiurewicz, Horseshoes, entropy and periods for graph maps, Topology, 32 (1993), 649-664.
doi: 10.1016/0040-9383(93)90014-M. |
[27] |
M. Mazur, Tolerance stability conjecture revisited, Topology Appl., 131 (2003), 33-38.
doi: 10.1016/S0166-8641(02)00261-4. |
[28] |
M. Mazur and P. Oprocha, S-limit shadowing is $C^0$-dense, J. Math. Anal. Appl., 408 (2013), 465-475.
doi: 10.1016/j.jmaa.2013.06.004. |
[29] |
I. Mizera, Generic properties of one-dimensional dynamical systems, in Ergodic Theory and Reletad Topics III (eds. U. Krengel, K. Richter and V. Warstat), Springer, (1992), 163-173.
doi: 10.1007/BFb0097537. |
[30] |
E. E. Moise, Geometric Topology in Dimensions $2$ and $3$, Springer-Verlag, New York-Heidelberg, 1977. |
[31] |
K. Odani, Generic homeomorphisms have the pseudo-orbit tracing property, Proc. Amer. Math. Soc., 110 (1990), 281-284.
doi: 10.1090/S0002-9939-1990-1009998-8. |
[32] |
A. V. Osipov, S. Yu. Pilyugin and S. B. Tikhomirov, Periodic shadowing and $\Omega$-stability, Regul. Chaotic Dyn., 15 (2010), 404-417.
doi: 10.1134/S1560354710020255. |
[33] |
K. Palmer, Shadowing in Dynamical Systems. Theory and Applications, Kluwer Academic Publishers, Dordrecht, 2000. |
[34] |
S. Yu. Pilyugin, Shadowing in Dynamical Systems, Springer-Verlag, Berlin, 1999. |
[35] |
S. Yu. Pilyugin, Inverse shadowing by continuous methods, Discrete Contin. Dyn. Syst., 8 (2002), 29-38.
doi: 10.3934/dcds.2002.8.29. |
[36] |
S. Yu. Pilyugin and O. B. Plamenevskaya, Shadowing is generic, Topology Appl., 97 (1999), 253-266.
doi: 10.1016/S0166-8641(98)00062-5. |
[37] |
S. Yu. Pilyugin, A. A. Rodionova and K. Sakai, Orbital and weak shadowing properties, Discrete Contin. Dyn. Syst., 9 (2003), 287-308.
doi: 10.3934/dcds.2003.9.287. |
[38] |
K. Sakai, Diffeomorphisms with pseudo-orbit tracing property, Nagoya Math. J., 126 (1992), 125-140. |
[39] |
K. Sakai, Pseudo-orbit tracing property and strong transversality of diffeomorphisms on closed manifolds, Osaka J. Math., 31 (1994), 373-386. |
[40] |
K. Sakai, Diffeomorphisms with the s-limit shadowing property, Dyn. Syst., 27 (2012), 403-410.
doi: 10.1080/14689367.2012.691960. |
[41] |
Y. Shi and Q. Xing, Dense distribution of chaotic maps in continuous map spaces, Dyn. Syst., 26 (2011), 519-535.
doi: 10.1080/14689367.2011.627836. |
[42] |
J. H. C. Whitehead, On $C^1$-complexes, Ann. of Math., 41 (1940), 809-824.
doi: 10.2307/1968861. |
[43] |
K. Yano, Generic homeomorphisms of $S^1$ have the pseudo-orbit tracing property, J. Fac. Sci. Univ. Tokyo Sect. IA Math., 34 (1987), 51-55. |
[44] |
G.-C. Yuan, J.A. Yorke, An open set of maps for which every point is absolutely nonshadowable, Proc. Amer. Math. Soc., 128 (2000), 909-918.
doi: 10.1090/S0002-9939-99-05038-8. |
[45] |
P. Zgliczyński and M. Gidea, Covering relations for multidimensional dynamical systems, J. Differential Equations, 202 (2004), 32-58.
doi: 10.1016/j.jde.2004.03.013. |
show all references
References:
[1] |
F. Abdenur and L. Díaz, Pseudo-orbit shadowing in the $C^1$ topology, Discrete Contin. Dyn. Syst., 17 (2007), 223-245. |
[2] |
D. V. Anosov, Geodesic Flows on Closed Riemann Manifolds with Negative Curvature, (translated from the Russian by S. Feder), American Mathematical Society, Providence, R.I., 1969. |
[3] |
N. Aoki and K. Hiraide, Topological Theory of Dynamical Systems. Recent Advances, North-Holland Publishing Co., Amsterdam, 1994. |
[4] |
R. H. Bing, An alternative proof that $3$-manifolds can be triangulated, Ann. of Math., 69 (1959), 37-65.
doi: 10.2307/1970092. |
[5] |
C. Bonatti, L. J. Díaz and G. Turcat, Pas de "shadowing lemma'' pour des dynamiques partiellement hyperboliques (French) [There is no shadowing lemma for partially hyperbolic dynamics], C. R. Acad. Sci. Paris Sér. I Math., 330 (2000), 587-592.
doi: 10.1016/S0764-4442(00)00215-9. |
[6] |
R. Bowen, $\omega$-limit sets for Axiom A diffeomorphisms, J. Differential Equations, 18 (1975), 333-339.
doi: 10.1016/0022-0396(75)90065-0. |
[7] |
S. S. Cairns, Triangulation of the manifold of class one, Bull. Amer. Math. Soc., 41 (1935), 549-552.
doi: 10.1090/S0002-9904-1935-06140-3. |
[8] |
L. Chen, Linking and the shadowing property for piecewise monotone maps, Proc. Amer. Math. Soc., 113 (1991), 251-263.
doi: 10.1090/S0002-9939-1991-1079695-2. |
[9] |
B. A. Coomes, H. Koçak and K. J. Palmer, Periodic shadowing, in Chaotic numerics(eds. P.E. Kloeden and K.J. Palmer), Amer. Math. Soc., Providence, R.I., (1994), 115-130.
doi: 10.1090/conm/172/01801. |
[10] |
R. M. Corless and S. Yu. Pilyugin, Approximate and real trajectories for generic dynamical systems, J. Math. Anal. Appl., 189 (1995), 409-423.
doi: 10.1006/jmaa.1995.1027. |
[11] |
E. M. Coven, I. Kan and J. A. Yorke, Pseudo-orbit shadowing in the family of tent maps, Trans. Amer. Math. Soc., 308 (1988), 227-241.
doi: 10.1090/S0002-9947-1988-0946440-2. |
[12] |
S. Crovisier, Periodic orbits and chain-transitive sets of $C^1$-diffeomorphisms, Publ. Math. Inst. Hautes Études Sci., 104 (2006), 87-141.
doi: 10.1007/s10240-006-0002-4. |
[13] |
J. Dugundji, An extension of Tietze's theorem, Pacific J. Math., 1 (1951), 353-367.
doi: 10.2140/pjm.1951.1.353. |
[14] |
J. Franks and C. A. Robinson, A quasi-Anosov diffeomorphism that is not Anosov, Trans. Amer. Math. Soc., 223 (1976), 267-278.
doi: 10.1090/S0002-9947-1976-0423420-9. |
[15] |
M. H. Freedman, The topology of four-dimensional manifolds, J. Differential Geom., 17 (1982), 357-453. |
[16] |
R. C. Kirby and L. C. Siebenmann, On the triangulation of manifolds and the Hauptvermutung, Bull. Amer. Math. Soc., 75 (1969), 742-749.
doi: 10.1090/S0002-9904-1969-12271-8. |
[17] |
R. C. Kirby and L. C. Siebenmann, Foundational Essays on Topological Manifolds, Smoothings, and Triangulations, Princeton University Press, Princeton, N.J., 1977. |
[18] |
U. Kirchgraber, U. Manz and D. Stoffer, Rigorous proof of chaotic behaviour in a dumbbell satellite model, J. Math. Anal. Appl., 251 (2000), 897-911.
doi: 10.1006/jmaa.2000.7143. |
[19] |
U. Kirchgraber and D. Stoffer, Possible chaotic motion of comets in the Sun-Jupiter system-a computer-assisted approach based on shadowing, Nonlinearity, 17 (2004), 281-300.
doi: 10.1088/0951-7715/17/1/016. |
[20] |
P. E. Kloeden and J. Ombach, Hyperbolic homeomorphisms are bishadowing, Ann. Polon. Math., 65 (1997), 171-177. |
[21] |
P. Kościelniak, On genericity of shadowing and periodic shadowing property, J. Math. Anal. Appl., 310 (2005), 188-196.
doi: 10.1016/j.jmaa.2005.01.053. |
[22] |
P. Kościelniak, Generic properties of $\mathbb Z^{2}$-actions on the interval, Topology Appl., 154 (2007), 2672-2677.
doi: 10.1016/j.topol.2007.05.001. |
[23] |
P. Kościelniak and M. Mazur, On $C^0$ genericity of various shadowing properties, Discrete Contin. Dyn. Syst., 12 (2005), 523-530. |
[24] |
P. Kościelniak and M. Mazur, Chaos and the shadowing property, Topology Appl., 154 (2007), 2553-2557.
doi: 10.1016/j.topol.2006.06.010. |
[25] |
P. Kościelniak and M. Mazur, Genericity of inverse shadowing property, J. Difference Equ. Appl., 16 (2010), 667-674.
doi: 10.1080/10236190903213464. |
[26] |
J. Llibre and M. Misiurewicz, Horseshoes, entropy and periods for graph maps, Topology, 32 (1993), 649-664.
doi: 10.1016/0040-9383(93)90014-M. |
[27] |
M. Mazur, Tolerance stability conjecture revisited, Topology Appl., 131 (2003), 33-38.
doi: 10.1016/S0166-8641(02)00261-4. |
[28] |
M. Mazur and P. Oprocha, S-limit shadowing is $C^0$-dense, J. Math. Anal. Appl., 408 (2013), 465-475.
doi: 10.1016/j.jmaa.2013.06.004. |
[29] |
I. Mizera, Generic properties of one-dimensional dynamical systems, in Ergodic Theory and Reletad Topics III (eds. U. Krengel, K. Richter and V. Warstat), Springer, (1992), 163-173.
doi: 10.1007/BFb0097537. |
[30] |
E. E. Moise, Geometric Topology in Dimensions $2$ and $3$, Springer-Verlag, New York-Heidelberg, 1977. |
[31] |
K. Odani, Generic homeomorphisms have the pseudo-orbit tracing property, Proc. Amer. Math. Soc., 110 (1990), 281-284.
doi: 10.1090/S0002-9939-1990-1009998-8. |
[32] |
A. V. Osipov, S. Yu. Pilyugin and S. B. Tikhomirov, Periodic shadowing and $\Omega$-stability, Regul. Chaotic Dyn., 15 (2010), 404-417.
doi: 10.1134/S1560354710020255. |
[33] |
K. Palmer, Shadowing in Dynamical Systems. Theory and Applications, Kluwer Academic Publishers, Dordrecht, 2000. |
[34] |
S. Yu. Pilyugin, Shadowing in Dynamical Systems, Springer-Verlag, Berlin, 1999. |
[35] |
S. Yu. Pilyugin, Inverse shadowing by continuous methods, Discrete Contin. Dyn. Syst., 8 (2002), 29-38.
doi: 10.3934/dcds.2002.8.29. |
[36] |
S. Yu. Pilyugin and O. B. Plamenevskaya, Shadowing is generic, Topology Appl., 97 (1999), 253-266.
doi: 10.1016/S0166-8641(98)00062-5. |
[37] |
S. Yu. Pilyugin, A. A. Rodionova and K. Sakai, Orbital and weak shadowing properties, Discrete Contin. Dyn. Syst., 9 (2003), 287-308.
doi: 10.3934/dcds.2003.9.287. |
[38] |
K. Sakai, Diffeomorphisms with pseudo-orbit tracing property, Nagoya Math. J., 126 (1992), 125-140. |
[39] |
K. Sakai, Pseudo-orbit tracing property and strong transversality of diffeomorphisms on closed manifolds, Osaka J. Math., 31 (1994), 373-386. |
[40] |
K. Sakai, Diffeomorphisms with the s-limit shadowing property, Dyn. Syst., 27 (2012), 403-410.
doi: 10.1080/14689367.2012.691960. |
[41] |
Y. Shi and Q. Xing, Dense distribution of chaotic maps in continuous map spaces, Dyn. Syst., 26 (2011), 519-535.
doi: 10.1080/14689367.2011.627836. |
[42] |
J. H. C. Whitehead, On $C^1$-complexes, Ann. of Math., 41 (1940), 809-824.
doi: 10.2307/1968861. |
[43] |
K. Yano, Generic homeomorphisms of $S^1$ have the pseudo-orbit tracing property, J. Fac. Sci. Univ. Tokyo Sect. IA Math., 34 (1987), 51-55. |
[44] |
G.-C. Yuan, J.A. Yorke, An open set of maps for which every point is absolutely nonshadowable, Proc. Amer. Math. Soc., 128 (2000), 909-918.
doi: 10.1090/S0002-9939-99-05038-8. |
[45] |
P. Zgliczyński and M. Gidea, Covering relations for multidimensional dynamical systems, J. Differential Equations, 202 (2004), 32-58.
doi: 10.1016/j.jde.2004.03.013. |
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