Shadowing is generic---a continuous map case

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September 2014, 34(9): 3591-3609. doi: 10.3934/dcds.2014.34.3591

Shadowing is generic---a continuous map case

Piotr Kościelniak ^1, , Marcin Mazur ^1, , Piotr Oprocha ^2, and Paweł Pilarczyk ^3,

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

Received January 2013 Revised December 2013 Published March 2014

Abstract
Related Papers

We prove that shadowing (the pseudo-orbit tracing property), periodic shadowing (tracing periodic pseudo-orbits with periodic real trajectories), and inverse shadowing with respect to certain families of methods (tracing all real orbits of the system with pseudo-orbits generated by arbitrary methods from these families) are all generic in the class of continuous maps and in the class of continuous onto maps on compact topological manifolds (with or without boundary) that admit a decomposition (including triangulable manifolds and manifolds with handlebody).

Keywords: inverse shadowing, semidynamical system, Shadowing, manifold, pseudotrajectory, periodic shadowing, $C^{0}$ topology., continuous surjection, genericity, continuous map.

Mathematics Subject Classification: 37C50, 37C20, 54H20, 37B9.

Citation: Piotr Kościelniak, Marcin Mazur, Piotr Oprocha, Paweł Pilarczyk. Shadowing is generic---a continuous map case. Discrete and Continuous Dynamical Systems, 2014, 34 (9) : 3591-3609. doi: 10.3934/dcds.2014.34.3591

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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[2]	S. Yu. Pilyugin. Inverse shadowing by continuous methods. Discrete and Continuous Dynamical Systems, 2002, 8 (1) : 29-38. doi: 10.3934/dcds.2002.8.29
[3]	Flavio Abdenur, Lorenzo J. Díaz. Pseudo-orbit shadowing in the $C^1$ topology. Discrete and Continuous Dynamical Systems, 2007, 17 (2) : 223-245. doi: 10.3934/dcds.2007.17.223
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