Shadowing is generic---a continuous map case

Piotr Kościelniak; Marcin Mazur; Piotr Oprocha; Paweł Pilarczyk

doi:10.3934/dcds.2014.34.3591

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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
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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 & 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. Google Scholar
[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. Google Scholar
[3]	N. Aoki and K. Hiraide, Topological Theory of Dynamical Systems. Recent Advances, North-Holland Publishing Co., Amsterdam, 1994. Google Scholar
[4]	R. H. Bing, An alternative proof that $3$-manifolds can be triangulated, Ann. of Math., 69 (1959), 37-65. doi: 10.2307/1970092. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[13]	J. Dugundji, An extension of Tietze's theorem, Pacific J. Math., 1 (1951), 353-367. doi: 10.2140/pjm.1951.1.353. Google Scholar
[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. Google Scholar
[15]	M. H. Freedman, The topology of four-dimensional manifolds, J. Differential Geom., 17 (1982), 357-453. Google Scholar
[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. Google Scholar
[17]	R. C. Kirby and L. C. Siebenmann, Foundational Essays on Topological Manifolds, Smoothings, and Triangulations, Princeton University Press, Princeton, N.J., 1977. Google Scholar
[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. Google Scholar
[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. Google Scholar
[20]	P. E. Kloeden and J. Ombach, Hyperbolic homeomorphisms are bishadowing, Ann. Polon. Math., 65 (1997), 171-177. Google Scholar
[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. Google Scholar
[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. Google Scholar
[23]	P. Kościelniak and M. Mazur, On $C^0$ genericity of various shadowing properties, Discrete Contin. Dyn. Syst., 12 (2005), 523-530. Google Scholar
[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. Google Scholar
[25]	P. Kościelniak and M. Mazur, Genericity of inverse shadowing property, J. Difference Equ. Appl., 16 (2010), 667-674. doi: 10.1080/10236190903213464. Google Scholar
[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. Google Scholar
[27]	M. Mazur, Tolerance stability conjecture revisited, Topology Appl., 131 (2003), 33-38. doi: 10.1016/S0166-8641(02)00261-4. Google Scholar
[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. Google Scholar
[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. Google Scholar
[30]	E. E. Moise, Geometric Topology in Dimensions $2$ and $3$, Springer-Verlag, New York-Heidelberg, 1977. Google Scholar
[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. Google Scholar
[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. Google Scholar
[33]	K. Palmer, Shadowing in Dynamical Systems. Theory and Applications, Kluwer Academic Publishers, Dordrecht, 2000. Google Scholar
[34]	S. Yu. Pilyugin, Shadowing in Dynamical Systems, Springer-Verlag, Berlin, 1999. Google Scholar
[35]	S. Yu. Pilyugin, Inverse shadowing by continuous methods, Discrete Contin. Dyn. Syst., 8 (2002), 29-38. doi: 10.3934/dcds.2002.8.29. Google Scholar
[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. Google Scholar
[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. Google Scholar
[38]	K. Sakai, Diffeomorphisms with pseudo-orbit tracing property, Nagoya Math. J., 126 (1992), 125-140. Google Scholar
[39]	K. Sakai, Pseudo-orbit tracing property and strong transversality of diffeomorphisms on closed manifolds, Osaka J. Math., 31 (1994), 373-386. Google Scholar
[40]	K. Sakai, Diffeomorphisms with the s-limit shadowing property, Dyn. Syst., 27 (2012), 403-410. doi: 10.1080/14689367.2012.691960. Google Scholar
[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. Google Scholar
[42]	J. H. C. Whitehead, On $C^1$-complexes, Ann. of Math., 41 (1940), 809-824. doi: 10.2307/1968861. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar

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. Google Scholar
[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. Google Scholar
[3]	N. Aoki and K. Hiraide, Topological Theory of Dynamical Systems. Recent Advances, North-Holland Publishing Co., Amsterdam, 1994. Google Scholar
[4]	R. H. Bing, An alternative proof that $3$-manifolds can be triangulated, Ann. of Math., 69 (1959), 37-65. doi: 10.2307/1970092. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[13]	J. Dugundji, An extension of Tietze's theorem, Pacific J. Math., 1 (1951), 353-367. doi: 10.2140/pjm.1951.1.353. Google Scholar
[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. Google Scholar
[15]	M. H. Freedman, The topology of four-dimensional manifolds, J. Differential Geom., 17 (1982), 357-453. Google Scholar
[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. Google Scholar
[17]	R. C. Kirby and L. C. Siebenmann, Foundational Essays on Topological Manifolds, Smoothings, and Triangulations, Princeton University Press, Princeton, N.J., 1977. Google Scholar
[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. Google Scholar
[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. Google Scholar
[20]	P. E. Kloeden and J. Ombach, Hyperbolic homeomorphisms are bishadowing, Ann. Polon. Math., 65 (1997), 171-177. Google Scholar
[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. Google Scholar
[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. Google Scholar
[23]	P. Kościelniak and M. Mazur, On $C^0$ genericity of various shadowing properties, Discrete Contin. Dyn. Syst., 12 (2005), 523-530. Google Scholar
[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. Google Scholar
[25]	P. Kościelniak and M. Mazur, Genericity of inverse shadowing property, J. Difference Equ. Appl., 16 (2010), 667-674. doi: 10.1080/10236190903213464. Google Scholar
[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. Google Scholar
[27]	M. Mazur, Tolerance stability conjecture revisited, Topology Appl., 131 (2003), 33-38. doi: 10.1016/S0166-8641(02)00261-4. Google Scholar
[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. Google Scholar
[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. Google Scholar
[30]	E. E. Moise, Geometric Topology in Dimensions $2$ and $3$, Springer-Verlag, New York-Heidelberg, 1977. Google Scholar
[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. Google Scholar
[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. Google Scholar
[33]	K. Palmer, Shadowing in Dynamical Systems. Theory and Applications, Kluwer Academic Publishers, Dordrecht, 2000. Google Scholar
[34]	S. Yu. Pilyugin, Shadowing in Dynamical Systems, Springer-Verlag, Berlin, 1999. Google Scholar
[35]	S. Yu. Pilyugin, Inverse shadowing by continuous methods, Discrete Contin. Dyn. Syst., 8 (2002), 29-38. doi: 10.3934/dcds.2002.8.29. Google Scholar
[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. Google Scholar
[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. Google Scholar
[38]	K. Sakai, Diffeomorphisms with pseudo-orbit tracing property, Nagoya Math. J., 126 (1992), 125-140. Google Scholar
[39]	K. Sakai, Pseudo-orbit tracing property and strong transversality of diffeomorphisms on closed manifolds, Osaka J. Math., 31 (1994), 373-386. Google Scholar
[40]	K. Sakai, Diffeomorphisms with the s-limit shadowing property, Dyn. Syst., 27 (2012), 403-410. doi: 10.1080/14689367.2012.691960. Google Scholar
[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. Google Scholar
[42]	J. H. C. Whitehead, On $C^1$-complexes, Ann. of Math., 41 (1940), 809-824. doi: 10.2307/1968861. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar

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