American Institute of Mathematical Sciences

Advanced
July  2017, 22(5): 1965-1975. doi: 10.3934/dcdsb.2017115

Topological stability in set-valued dynamics

Roger Metzger 1, , Carlos Arnoldo Morales Rojas 2,, and  Phillipe Thieullen 3,

1. 

Instituto de Matemàtica y Ciencias Afines (IMCA), Universidad Nacional de Ingeniera Calle Los Biòlogos 245, 15012 Lima, Perù
2. 

Instituto de Matemática, Universidade Federal do Rio de Janeiro, P. O. Box 68530,21945-970 Rio de Janeiro, Brazil
3. 

Institut de Mathématiques Université de Bordeaux Ⅰ, 33405, Talence, France

* Corresponding author: Carlos Arnoldo Morales Rojas

Received  November 2016 Revised  January 2017 Published  March 2017

Fund Project: Work partially supported by CNPq from Brazil and MATHAMSUD 15 MATH05-ERGOPTIM, Ergodic Optimization of Lyapunov Exponents.

We propose a definition of topological stability for set-valued maps. We prove that a single-valued map which is topologically stable in the set-valued sense is topologically stable in the classical sense [14]. Next, we prove that every upper semicontinuous closed-valued map which is positively expansive [15] and satisfies the positive pseudo-orbit tracing property [9] is topologically stable. Finally, we prove that every topologically stable set-valued map of a compact metric space has the positive pseudo-orbit tracing property and the periodic points are dense in the nonwandering set. These results extend the classical single-valued ones in [1] and [14].

Keywords: Topological stability, set-valued map, metric space.
Mathematics Subject Classification: Primary: 54H20; Secondary: 54C6.
Citation: Roger Metzger, Carlos Arnoldo Morales Rojas, Phillipe Thieullen. Topological stability in set-valued dynamics. Discrete & Continuous Dynamical Systems - B, 2017, 22 (5) : 1965-1975. doi: 10.3934/dcdsb.2017115
References:
[1]

N. Aoki and K. Hiraide, Topological Theory of Dynamical Systems. Recent Advances, North-Holland Mathematical Library, 52. North-Holland Publishing Co. , Amsterdam, 1994. Google Scholar

[2]

J. -P. Aubin and H. Frankowska, Set-valued Analysis, Reprint of the 1990 edition. Modern Birkhäuser Classics. Birkhäuser Boston, Inc. , Boston, MA, 2009. Google Scholar

[3]

R. Bowen, ω-limit sets for axiom A diffeomorphisms, J. Differential Equations, 18 (1975), 333-339.  doi: 10.1016/0022-0396(75)90065-0.  Google Scholar

[4]

D. Carrasco-Olivera, A. R. Metzger and C. A. Morales, Logarithmic expansion, entropy and dimension for set-valued maps, Preprint, (2016), to appear. Google Scholar

[5]

D. Carrasco-Olivera, R. Metzger Alvan and C. A. Morales, Topological entropy for set-valued maps, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), 3461-3474.  doi: 10.3934/dcdsb.2015.20.3461.  Google Scholar

[6]

W. Cordeiro and M. J. Pacifico, Continuum-wise expansiveness and specification for set-valued functions and topological entropy, Proc. Amer. Math. Soc., 144 (2016), 4261-4271.  doi: 10.1090/proc/13168.  Google Scholar

[7]

M. Eisenberg, Expansive transformation semigroups of endomorphisms, Fund. Math., 59 (1966), 313-321.   Google Scholar

[8]

J. P. Kelly and T. Tennant, Topological entropy for set-valued functions, arXiv: 1509.08413. Google Scholar

[9]

S. Y. Pilyugin and J. Rieger, Shadowing and inverse shadowing in set-valued dynamical systems. Contractive case, Topol. Methods Nonlinear Anal., 32 (2008), 139-149.   Google Scholar

[10]

S. Y. Pilyugin, Shadowing in Dynamical Systems Lecture Notes in Mathematics, 1706. Springer-Verlag, Berlin, 1999. Google Scholar

[11]

B. E. Raines and T. Tennant, The specification property on a set-valued map and its inverse limit, arXiv: 1509.08415. Google Scholar

[12]

W. R. Utz, Unstable homeomorphisms, Proc. Amer. Math. Soc., 1 (1950), 769-774.  doi: 10.1090/S0002-9939-1950-0038022-3.  Google Scholar

[13]

P. Walters, On the pseudo-orbit tracing property and its relationship to stability. The structure of attractors in dynamical systems (Proc. Conf. , North Dakota State Univ. , Fargo, N. D. , 1977), Lecture Notes in Math. , Springer, Berlin, 668 (1978), 231-244. Google Scholar

[14]

P. Walters, Anosov diffeomorphisms are topologically stable, Topology, 9 (1970), 71-78.  doi: 10.1016/0040-9383(70)90051-0.  Google Scholar

[15]

R. K. Williams, A note on expansive mappings, Proc. Amer. Math. Soc., 22 (1969), 145-147.  doi: 10.1090/S0002-9939-1969-0242143-4.  Google Scholar

show all references

References:
[1]

N. Aoki and K. Hiraide, Topological Theory of Dynamical Systems. Recent Advances, North-Holland Mathematical Library, 52. North-Holland Publishing Co. , Amsterdam, 1994. Google Scholar

[2]

J. -P. Aubin and H. Frankowska, Set-valued Analysis, Reprint of the 1990 edition. Modern Birkhäuser Classics. Birkhäuser Boston, Inc. , Boston, MA, 2009. Google Scholar

[3]

R. Bowen, ω-limit sets for axiom A diffeomorphisms, J. Differential Equations, 18 (1975), 333-339.  doi: 10.1016/0022-0396(75)90065-0.  Google Scholar

[4]

D. Carrasco-Olivera, A. R. Metzger and C. A. Morales, Logarithmic expansion, entropy and dimension for set-valued maps, Preprint, (2016), to appear. Google Scholar

[5]

D. Carrasco-Olivera, R. Metzger Alvan and C. A. Morales, Topological entropy for set-valued maps, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), 3461-3474.  doi: 10.3934/dcdsb.2015.20.3461.  Google Scholar

[6]

W. Cordeiro and M. J. Pacifico, Continuum-wise expansiveness and specification for set-valued functions and topological entropy, Proc. Amer. Math. Soc., 144 (2016), 4261-4271.  doi: 10.1090/proc/13168.  Google Scholar

[7]

M. Eisenberg, Expansive transformation semigroups of endomorphisms, Fund. Math., 59 (1966), 313-321.   Google Scholar

[8]

J. P. Kelly and T. Tennant, Topological entropy for set-valued functions, arXiv: 1509.08413. Google Scholar

[9]

S. Y. Pilyugin and J. Rieger, Shadowing and inverse shadowing in set-valued dynamical systems. Contractive case, Topol. Methods Nonlinear Anal., 32 (2008), 139-149.   Google Scholar

[10]

S. Y. Pilyugin, Shadowing in Dynamical Systems Lecture Notes in Mathematics, 1706. Springer-Verlag, Berlin, 1999. Google Scholar

[11]

B. E. Raines and T. Tennant, The specification property on a set-valued map and its inverse limit, arXiv: 1509.08415. Google Scholar

[12]

W. R. Utz, Unstable homeomorphisms, Proc. Amer. Math. Soc., 1 (1950), 769-774.  doi: 10.1090/S0002-9939-1950-0038022-3.  Google Scholar

[13]

P. Walters, On the pseudo-orbit tracing property and its relationship to stability. The structure of attractors in dynamical systems (Proc. Conf. , North Dakota State Univ. , Fargo, N. D. , 1977), Lecture Notes in Math. , Springer, Berlin, 668 (1978), 231-244. Google Scholar

[14]

P. Walters, Anosov diffeomorphisms are topologically stable, Topology, 9 (1970), 71-78.  doi: 10.1016/0040-9383(70)90051-0.  Google Scholar

[15]

R. K. Williams, A note on expansive mappings, Proc. Amer. Math. Soc., 22 (1969), 145-147.  doi: 10.1090/S0002-9939-1969-0242143-4.  Google Scholar

[1]

Shasha Hu, Yihong Xu, Yuhan Zhang. Second-Order characterizations for set-valued equilibrium problems with variable ordering structures. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020164
[2]

Vaibhav Mehandiratta, Mani Mehra, Günter Leugering. Existence results and stability analysis for a nonlinear fractional boundary value problem on a circular ring with an attached edge : A study of fractional calculus on metric graph. Networks & Heterogeneous Media, 2021  doi: 10.3934/nhm.2021003
[3]

Jingjing Wang, Zaiyun Peng, Zhi Lin, Daqiong Zhou. On the stability of solutions for the generalized vector quasi-equilibrium problems via free-disposal set. Journal of Industrial & Management Optimization, 2021, 17 (2) : 869-887. doi: 10.3934/jimo.2020002
[4]

Yunping Jiang. Global graph of metric entropy on expanding Blaschke products. Discrete & Continuous Dynamical Systems - A, 2021, 41 (3) : 1469-1482. doi: 10.3934/dcds.2020325
[5]

Pablo D. Carrasco, Túlio Vales. A symmetric Random Walk defined by the time-one map of a geodesic flow. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020390
[6]

Tuoc Phan, Grozdena Todorova, Borislav Yordanov. Existence uniqueness and regularity theory for elliptic equations with complex-valued potentials. Discrete & Continuous Dynamical Systems - A, 2021, 41 (3) : 1071-1099. doi: 10.3934/dcds.2020310
[7]

Tian Ma, Shouhong Wang. Topological phase transition III: Solar surface eruptions and sunspots. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 501-514. doi: 10.3934/dcdsb.2020350
[8]

Álvaro Castañeda, Pablo González, Gonzalo Robledo. Topological Equivalence of nonautonomous difference equations with a family of dichotomies on the half line. Communications on Pure & Applied Analysis, 2021, 20 (2) : 511-532. doi: 10.3934/cpaa.2020278
[9]

Giulia Luise, Giuseppe Savaré. Contraction and regularizing properties of heat flows in metric measure spaces. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 273-297. doi: 10.3934/dcdss.2020327
[10]

Soonki Hong, Seonhee Lim. Martin boundary of brownian motion on Gromov hyperbolic metric graphs. Discrete & Continuous Dynamical Systems - A, 2021  doi: 10.3934/dcds.2021014
[11]

Sergei Avdonin, Yuanyuan Zhao. Leaf Peeling method for the wave equation on metric tree graphs. Inverse Problems & Imaging, 2021, 15 (2) : 185-199. doi: 10.3934/ipi.2020060
[12]

Mostafa Mbekhta. Representation and approximation of the polar factor of an operator on a Hilbert space. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020463
[13]

Michiel Bertsch, Flavia Smarrazzo, Andrea Terracina, Alberto Tesei. Signed Radon measure-valued solutions of flux saturated scalar conservation laws. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3143-3169. doi: 10.3934/dcds.2020041
[14]

Simone Fiori. Error-based control systems on Riemannian state manifolds: Properties of the principal pushforward map associated to parallel transport. Mathematical Control & Related Fields, 2021, 11 (1) : 143-167. doi: 10.3934/mcrf.2020031
[15]

Jann-Long Chern, Sze-Guang Yang, Zhi-You Chen, Chih-Her Chen. On the family of non-topological solutions for the elliptic system arising from a product Abelian gauge field theory. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3291-3304. doi: 10.3934/dcds.2020127
[16]

Huu-Quang Nguyen, Ya-Chi Chu, Ruey-Lin Sheu. On the convexity for the range set of two quadratic functions. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020169
[17]

Sumit Kumar Debnath, Pantelimon Stǎnicǎ, Nibedita Kundu, Tanmay Choudhury. Secure and efficient multiparty private set intersection cardinality. Advances in Mathematics of Communications, 2021, 15 (2) : 365-386. doi: 10.3934/amc.2020071
[18]

Hirokazu Ninomiya. Entire solutions of the Allen–Cahn–Nagumo equation in a multi-dimensional space. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 395-412. doi: 10.3934/dcds.2020364
[19]

Dong-Ho Tsai, Chia-Hsing Nien. On space-time periodic solutions of the one-dimensional heat equation. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3997-4017. doi: 10.3934/dcds.2020037
[20]

Petr Čoupek, María J. Garrido-Atienza. Bilinear equations in Hilbert space driven by paths of low regularity. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 121-154. doi: 10.3934/dcdsb.2020230

2019 Impact Factor: 1.27

Tools

Article outline

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences