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December  2015, 20(10): 3461-3474. doi: 10.3934/dcdsb.2015.20.3461

Topological entropy for set-valued maps

Dante Carrasco-Olivera 1, , Roger Metzger Alvan 2, and  Carlos Arnoldo Morales Rojas 3,

1. 

Departamento de Matemática, Universidad del Bío, Bío Av. Collao # 1202, Casilla 5-C, VIII-Región, Concepción, Chile
2. 

Instituto de Matemática y Ciencias Afines (IMCA), Universidad Nacional de Ingeniería, Calle Los Biólogos 245, Urb. San César La Molina, Lima 12, Lima, Peru
3. 

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

Received  April 2015 Revised  May 2015 Published  September 2015

In this paper we define and study the topological entropy of a set-valued dynamical system. Actually, we obtain two entropies based on separated and spanning sets. Some properties of these entropies resembling the single-valued case will be obtained.
Keywords: spanning set, separated set., Set-valued map, metric space, topological entropy.
Mathematics Subject Classification: Primary: 37B40; Secondary: 54C6.
Citation: Dante Carrasco-Olivera, Roger Metzger Alvan, Carlos Arnoldo Morales Rojas. Topological entropy for set-valued maps. Discrete & Continuous Dynamical Systems - B, 2015, 20 (10) : 3461-3474. doi: 10.3934/dcdsb.2015.20.3461
References:
[1]

R. L. Adler, A. G. Konheim and M. H. McAndrew, Topological entropy, Trans. Amer. Math. Soc., 114 (1965), 309-319. doi: 10.1090/S0002-9947-1965-0175106-9.  Google Scholar

[2]

E. Akin, The General Topology of Dynamical Systems, Graduate Studies in Mathematics, 1, American Mathematical Society, Providence, RI, 1993.  Google Scholar

[3]

J.-P. Aubin and A. Cellina, Differential Inclusions. Set-Valued Maps and Viability Theory, Grundlehren der Mathematischen Wissenschaften, 264, Springer-Verlag, Berlin, 1984. doi: 10.1007/978-3-642-69512-4.  Google Scholar

[4]

J.-P. Aubin and H. Frankowska, Set-valued Analysis, Birkhäuser Boston, Inc., Boston, MA, 1990; Modern Birkhäuser Classics, Birkhüser Boston, Inc., Boston, MA, 2009.  Google Scholar

[5]

J.-P. Aubin, H. Frankowska and A. Lasota, Poincaré's recurrence theorem for set-valued dynamical systems, Ann. Polon. Math., 54 (1991), 85-91.  Google Scholar

[6]

L. M. Blumenthal, A new concept in distance geometry with applications to spherical subsets, Bull. Amer. Math. Soc., 47 (1941), 435-443. doi: 10.1090/S0002-9904-1941-07471-9.  Google Scholar

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L. Boltzmann, Über die beziehung dem zweiten Haubtsatze der mechanischen Wärmetheorie und der Wahrscheinlichkeitsrechnung respektive den Sätzen über das Wärmegleichgewicht, Wiener Berichte, 76 (1877), 373-435. Google Scholar

[8]

R. Bowen, Entropy for group endomorphisms and homogeneous spaces, Trans. Amer. Math. Soc., 153 (1971), 401-414. doi: 10.1090/S0002-9947-1971-0274707-X.  Google Scholar

[9]

M. Brin and G. Stuck, Introduction to Dynamical Systems, Cambridge University Press, Cambridge, 2002. doi: 10.1017/CBO9780511755316.  Google Scholar

[10]

R. S. Burachik and A. N. Iusem, Set-valued Mappings and Enlargements of Monotone Operators, Springer Optimization and Its Applications, 8, Springer, New York, 2008.  Google Scholar

[11]

L. J. Cherene, Jr., Set Valued Dynamical Systems and Economic Flow, Lecture Notes in Economics and Mathematical Systems, 158, Springer-Verlag, Berlin-New York, 1978.  Google Scholar

[12]

M. Ciklová, Dynamical systems generated by functions with connected $G_\delta$ graphs,, Real Anal. Exchange, 30 (): 617.   Google Scholar

[13]

R. Clausius, Ueber verschiedene für die Anwendung bequeme Formen der Hauptgleichungen der mechanischen Wärmetheorie,, Annalen der Physik, 125 (): 353.   Google Scholar

[14]

R. Clausius, The Mechanical Theory of Heat - with its Applications to the Steam Engine and to Physical Properties of Bodies, John van Voorst, London, 1867. Google Scholar

[15]

E. I. Dinaburg, A correlation between topological entropy and metric entropy (Russian), Dokl. Akad. Nauk SSSR, 190 (1970), 19-22.  Google Scholar

[16]

T. Downarowicz, Entropy in Dynamical Systems, New Mathematical Monographs, 18, Cambridge University Press, Cambridge, 2011. doi: 10.1017/CBO9780511976155.  Google Scholar

[17]

B. E. Gillam, A new set of postulates for euclidean geometry, Revista Ci. Lima, 42 (1940), 869-899.  Google Scholar

[18]

A. Katok, Fifty years of entropy in dynamics: 1958-2007, J. Mod. Dyn., 1 (2007), 545-596. doi: 10.3934/jmd.2007.1.545.  Google Scholar

[19]

A. Katok, Lyapunov exponents, entropy and periodic orbits for diffeomorphisms, Inst. Hautes Études Sci. Publ. Math., 51 (1980), 137-173.  Google Scholar

[20]

A. Y. Khinchin, On the basic theorems of information theory, Uspehi Mat. Nauk (N.S.), 11 (1956), 17-75.  Google Scholar

[21]

A. N. Kolmogorov, A new metric invariant of transitive dynamical systems and automorphisms of Lebesgue spaces, (Russian) Topology, ordinary differential equations, dynamical systems, Trudy Mat. Inst. Steklov., 169 (1985), 94-98, 254.  Google Scholar

[22]

M. Maschler and B. Peleg, Stable sets and stable points of set-valued dynamic systems with applications to game theory, SIAM J. Control Optimization, 14 (1976), 985-995. doi: 10.1137/0314062.  Google Scholar

[23]

B. McMillan, The basic theorems of information theory, Ann. Math. Statistics, 24 (1953), 196-219. doi: 10.1214/aoms/1177729028.  Google Scholar

[24]

W. M. Miller, Frobenius-Perron operators and approximation of invariant measures for set-valued dynamical systems, Set-Valued Anal., 3 (1995), 181-194. doi: 10.1007/BF01038599.  Google Scholar

[25]

W. Miller and E. Akin, Invariant measures for set-valued dynamical systems, Trans. Amer. Math. Soc., 351 (1999), 1203-1225. doi: 10.1090/S0002-9947-99-02424-1.  Google Scholar

[26]

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

[27]

S. Y. Pilyugin and J. Rieger, Shadowing and inverse shadowing in set-valued dynamical systems, Hyperbolic case, Topol. Methods Nonlinear Anal., 32 (2008), 151-164.  Google Scholar

[28]

R. T. Rockafellar, Convex Analysis, Reprint of the 1970 original, Princeton Landmarks in Mathematics, Princeton Paperbacks, Princeton University Press, Princeton, NJ, 1997.  Google Scholar

[29]

Ja. Sinai, On the concept of entropy for a dynamic system, (Russian) Dokl. Akad. Nauk SSSR, 124 (1959), 768-771.  Google Scholar

[30]

Y. Sinai, Kolmogorov-Sinai entropy, Scholarpedia, 4 (2009), p2034. Google Scholar

[31]

C. E. Shannon, A mathematical theory of communication, Bell System Tech. J., 27 (1948), 379-423, 623-656. doi: 10.1002/j.1538-7305.1948.tb01338.x.  Google Scholar

[32]

E. Tarafdar, P. Watson and X.-Z. Yuan, Poincare's recurrence theorems for set-valued dynamical systems, Appl. Math. Lett., 10 (1997), 37-44. doi: 10.1016/S0893-9659(97)00102-X.  Google Scholar

[33]

E. Tarafdar and X.-Z. Yuan, The set-valued dynamic system and its applications to Pareto optima, Acta Appl. Math., 46 (1997), 93-106. doi: 10.1023/A:1005722506504.  Google Scholar

[34]

J. von Neumann, Mathematische Grundlagen der Quantenmechanik, Unveränderter Nachdruck der ersten Auflage von 1932, Die Grundlehren der mathematischen Wissenschaften, Band 38, Springer-Verlag, Berlin-New York, 1968.  Google Scholar

[35]

P. Walters, An Introduction to Ergodic Theory, Graduate Texts in Mathematics, 79, Springer-Verlag, New York-Berlin, 1982.  Google Scholar

[36]

A. J. Zaslavski, Convergence of trajectories of discrete dispersive dynamical systems, Commun. Math. Anal., 4 (2008), 10-19.  Google Scholar

show all references

References:
[1]

R. L. Adler, A. G. Konheim and M. H. McAndrew, Topological entropy, Trans. Amer. Math. Soc., 114 (1965), 309-319. doi: 10.1090/S0002-9947-1965-0175106-9.  Google Scholar

[2]

E. Akin, The General Topology of Dynamical Systems, Graduate Studies in Mathematics, 1, American Mathematical Society, Providence, RI, 1993.  Google Scholar

[3]

J.-P. Aubin and A. Cellina, Differential Inclusions. Set-Valued Maps and Viability Theory, Grundlehren der Mathematischen Wissenschaften, 264, Springer-Verlag, Berlin, 1984. doi: 10.1007/978-3-642-69512-4.  Google Scholar

[4]

J.-P. Aubin and H. Frankowska, Set-valued Analysis, Birkhäuser Boston, Inc., Boston, MA, 1990; Modern Birkhäuser Classics, Birkhüser Boston, Inc., Boston, MA, 2009.  Google Scholar

[5]

J.-P. Aubin, H. Frankowska and A. Lasota, Poincaré's recurrence theorem for set-valued dynamical systems, Ann. Polon. Math., 54 (1991), 85-91.  Google Scholar

[6]

L. M. Blumenthal, A new concept in distance geometry with applications to spherical subsets, Bull. Amer. Math. Soc., 47 (1941), 435-443. doi: 10.1090/S0002-9904-1941-07471-9.  Google Scholar

[7]

L. Boltzmann, Über die beziehung dem zweiten Haubtsatze der mechanischen Wärmetheorie und der Wahrscheinlichkeitsrechnung respektive den Sätzen über das Wärmegleichgewicht, Wiener Berichte, 76 (1877), 373-435. Google Scholar

[8]

R. Bowen, Entropy for group endomorphisms and homogeneous spaces, Trans. Amer. Math. Soc., 153 (1971), 401-414. doi: 10.1090/S0002-9947-1971-0274707-X.  Google Scholar

[9]

M. Brin and G. Stuck, Introduction to Dynamical Systems, Cambridge University Press, Cambridge, 2002. doi: 10.1017/CBO9780511755316.  Google Scholar

[10]

R. S. Burachik and A. N. Iusem, Set-valued Mappings and Enlargements of Monotone Operators, Springer Optimization and Its Applications, 8, Springer, New York, 2008.  Google Scholar

[11]

L. J. Cherene, Jr., Set Valued Dynamical Systems and Economic Flow, Lecture Notes in Economics and Mathematical Systems, 158, Springer-Verlag, Berlin-New York, 1978.  Google Scholar

[12]

M. Ciklová, Dynamical systems generated by functions with connected $G_\delta$ graphs,, Real Anal. Exchange, 30 (): 617.   Google Scholar

[13]

R. Clausius, Ueber verschiedene für die Anwendung bequeme Formen der Hauptgleichungen der mechanischen Wärmetheorie,, Annalen der Physik, 125 (): 353.   Google Scholar

[14]

R. Clausius, The Mechanical Theory of Heat - with its Applications to the Steam Engine and to Physical Properties of Bodies, John van Voorst, London, 1867. Google Scholar

[15]

E. I. Dinaburg, A correlation between topological entropy and metric entropy (Russian), Dokl. Akad. Nauk SSSR, 190 (1970), 19-22.  Google Scholar

[16]

T. Downarowicz, Entropy in Dynamical Systems, New Mathematical Monographs, 18, Cambridge University Press, Cambridge, 2011. doi: 10.1017/CBO9780511976155.  Google Scholar

[17]

B. E. Gillam, A new set of postulates for euclidean geometry, Revista Ci. Lima, 42 (1940), 869-899.  Google Scholar

[18]

A. Katok, Fifty years of entropy in dynamics: 1958-2007, J. Mod. Dyn., 1 (2007), 545-596. doi: 10.3934/jmd.2007.1.545.  Google Scholar

[19]

A. Katok, Lyapunov exponents, entropy and periodic orbits for diffeomorphisms, Inst. Hautes Études Sci. Publ. Math., 51 (1980), 137-173.  Google Scholar

[20]

A. Y. Khinchin, On the basic theorems of information theory, Uspehi Mat. Nauk (N.S.), 11 (1956), 17-75.  Google Scholar

[21]

A. N. Kolmogorov, A new metric invariant of transitive dynamical systems and automorphisms of Lebesgue spaces, (Russian) Topology, ordinary differential equations, dynamical systems, Trudy Mat. Inst. Steklov., 169 (1985), 94-98, 254.  Google Scholar

[22]

M. Maschler and B. Peleg, Stable sets and stable points of set-valued dynamic systems with applications to game theory, SIAM J. Control Optimization, 14 (1976), 985-995. doi: 10.1137/0314062.  Google Scholar

[23]

B. McMillan, The basic theorems of information theory, Ann. Math. Statistics, 24 (1953), 196-219. doi: 10.1214/aoms/1177729028.  Google Scholar

[24]

W. M. Miller, Frobenius-Perron operators and approximation of invariant measures for set-valued dynamical systems, Set-Valued Anal., 3 (1995), 181-194. doi: 10.1007/BF01038599.  Google Scholar

[25]

W. Miller and E. Akin, Invariant measures for set-valued dynamical systems, Trans. Amer. Math. Soc., 351 (1999), 1203-1225. doi: 10.1090/S0002-9947-99-02424-1.  Google Scholar

[26]

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

[27]

S. Y. Pilyugin and J. Rieger, Shadowing and inverse shadowing in set-valued dynamical systems, Hyperbolic case, Topol. Methods Nonlinear Anal., 32 (2008), 151-164.  Google Scholar

[28]

R. T. Rockafellar, Convex Analysis, Reprint of the 1970 original, Princeton Landmarks in Mathematics, Princeton Paperbacks, Princeton University Press, Princeton, NJ, 1997.  Google Scholar

[29]

Ja. Sinai, On the concept of entropy for a dynamic system, (Russian) Dokl. Akad. Nauk SSSR, 124 (1959), 768-771.  Google Scholar

[30]

Y. Sinai, Kolmogorov-Sinai entropy, Scholarpedia, 4 (2009), p2034. Google Scholar

[31]

C. E. Shannon, A mathematical theory of communication, Bell System Tech. J., 27 (1948), 379-423, 623-656. doi: 10.1002/j.1538-7305.1948.tb01338.x.  Google Scholar

[32]

E. Tarafdar, P. Watson and X.-Z. Yuan, Poincare's recurrence theorems for set-valued dynamical systems, Appl. Math. Lett., 10 (1997), 37-44. doi: 10.1016/S0893-9659(97)00102-X.  Google Scholar

[33]

E. Tarafdar and X.-Z. Yuan, The set-valued dynamic system and its applications to Pareto optima, Acta Appl. Math., 46 (1997), 93-106. doi: 10.1023/A:1005722506504.  Google Scholar

[34]

J. von Neumann, Mathematische Grundlagen der Quantenmechanik, Unveränderter Nachdruck der ersten Auflage von 1932, Die Grundlehren der mathematischen Wissenschaften, Band 38, Springer-Verlag, Berlin-New York, 1968.  Google Scholar

[35]

P. Walters, An Introduction to Ergodic Theory, Graduate Texts in Mathematics, 79, Springer-Verlag, New York-Berlin, 1982.  Google Scholar

[36]

A. J. Zaslavski, Convergence of trajectories of discrete dispersive dynamical systems, Commun. Math. Anal., 4 (2008), 10-19.  Google Scholar

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