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Realizing subexponential entropy growth rates by cutting and stacking
Topological entropy for set-valued maps
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 |
References:
[1] |
R. L. Adler, A. G. Konheim and M. H. McAndrew, Topological entropy,, Trans. Amer. Math. Soc., 114 (1965), 309.
doi: 10.1090/S0002-9947-1965-0175106-9. |
[2] |
E. Akin, The General Topology of Dynamical Systems,, Graduate Studies in Mathematics, (1993).
|
[3] |
J.-P. Aubin and A. Cellina, Differential Inclusions. Set-Valued Maps and Viability Theory,, Grundlehren der Mathematischen Wissenschaften, (1984).
doi: 10.1007/978-3-642-69512-4. |
[4] |
J.-P. Aubin and H. Frankowska, Set-valued Analysis,, Birkhäuser Boston, (1990).
|
[5] |
J.-P. Aubin, H. Frankowska and A. Lasota, Poincaré's recurrence theorem for set-valued dynamical systems,, Ann. Polon. Math., 54 (1991), 85.
|
[6] |
L. M. Blumenthal, A new concept in distance geometry with applications to spherical subsets,, Bull. Amer. Math. Soc., 47 (1941), 435.
doi: 10.1090/S0002-9904-1941-07471-9. |
[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. Google Scholar |
[8] |
R. Bowen, Entropy for group endomorphisms and homogeneous spaces,, Trans. Amer. Math. Soc., 153 (1971), 401.
doi: 10.1090/S0002-9947-1971-0274707-X. |
[9] |
M. Brin and G. Stuck, Introduction to Dynamical Systems,, Cambridge University Press, (2002).
doi: 10.1017/CBO9780511755316. |
[10] |
R. S. Burachik and A. N. Iusem, Set-valued Mappings and Enlargements of Monotone Operators,, Springer Optimization and Its Applications, (2008).
|
[11] |
L. J. Cherene, Jr., Set Valued Dynamical Systems and Economic Flow,, Lecture Notes in Economics and Mathematical Systems, (1978).
|
[12] |
M. Ciklová, Dynamical systems generated by functions with connected $G_\delta$ graphs,, Real Anal. Exchange, 30 (): 617.
|
[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, (1867). Google Scholar |
[15] |
E. I. Dinaburg, A correlation between topological entropy and metric entropy (Russian),, Dokl. Akad. Nauk SSSR, 190 (1970), 19.
|
[16] |
T. Downarowicz, Entropy in Dynamical Systems,, New Mathematical Monographs, (2011).
doi: 10.1017/CBO9780511976155. |
[17] |
B. E. Gillam, A new set of postulates for euclidean geometry,, Revista Ci. Lima, 42 (1940), 869.
|
[18] |
A. Katok, Fifty years of entropy in dynamics: 1958-2007,, J. Mod. Dyn., 1 (2007), 545.
doi: 10.3934/jmd.2007.1.545. |
[19] |
A. Katok, Lyapunov exponents, entropy and periodic orbits for diffeomorphisms,, Inst. Hautes Études Sci. Publ. Math., 51 (1980), 137.
|
[20] |
A. Y. Khinchin, On the basic theorems of information theory,, Uspehi Mat. Nauk (N.S.), 11 (1956), 17.
|
[21] |
A. N. Kolmogorov, A new metric invariant of transitive dynamical systems and automorphisms of Lebesgue spaces, (Russian), Topology, 169 (1985), 94.
|
[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.
doi: 10.1137/0314062. |
[23] |
B. McMillan, The basic theorems of information theory,, Ann. Math. Statistics, 24 (1953), 196.
doi: 10.1214/aoms/1177729028. |
[24] |
W. M. Miller, Frobenius-Perron operators and approximation of invariant measures for set-valued dynamical systems,, Set-Valued Anal., 3 (1995), 181.
doi: 10.1007/BF01038599. |
[25] |
W. Miller and E. Akin, Invariant measures for set-valued dynamical systems,, Trans. Amer. Math. Soc., 351 (1999), 1203.
doi: 10.1090/S0002-9947-99-02424-1. |
[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.
|
[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.
|
[28] |
R. T. Rockafellar, Convex Analysis,, Reprint of the 1970 original, (1970).
|
[29] |
Ja. Sinai, On the concept of entropy for a dynamic system, (Russian), Dokl. Akad. Nauk SSSR, 124 (1959), 768.
|
[30] |
Y. Sinai, Kolmogorov-Sinai entropy,, Scholarpedia, 4 (2009). Google Scholar |
[31] |
C. E. Shannon, A mathematical theory of communication,, Bell System Tech. J., 27 (1948), 379.
doi: 10.1002/j.1538-7305.1948.tb01338.x. |
[32] |
E. Tarafdar, P. Watson and X.-Z. Yuan, Poincare's recurrence theorems for set-valued dynamical systems,, Appl. Math. Lett., 10 (1997), 37.
doi: 10.1016/S0893-9659(97)00102-X. |
[33] |
E. Tarafdar and X.-Z. Yuan, The set-valued dynamic system and its applications to Pareto optima,, Acta Appl. Math., 46 (1997), 93.
doi: 10.1023/A:1005722506504. |
[34] |
J. von Neumann, Mathematische Grundlagen der Quantenmechanik,, Unveränderter Nachdruck der ersten Auflage von 1932, (1932).
|
[35] |
P. Walters, An Introduction to Ergodic Theory,, Graduate Texts in Mathematics, (1982).
|
[36] |
A. J. Zaslavski, Convergence of trajectories of discrete dispersive dynamical systems,, Commun. Math. Anal., 4 (2008), 10.
|
show all references
References:
[1] |
R. L. Adler, A. G. Konheim and M. H. McAndrew, Topological entropy,, Trans. Amer. Math. Soc., 114 (1965), 309.
doi: 10.1090/S0002-9947-1965-0175106-9. |
[2] |
E. Akin, The General Topology of Dynamical Systems,, Graduate Studies in Mathematics, (1993).
|
[3] |
J.-P. Aubin and A. Cellina, Differential Inclusions. Set-Valued Maps and Viability Theory,, Grundlehren der Mathematischen Wissenschaften, (1984).
doi: 10.1007/978-3-642-69512-4. |
[4] |
J.-P. Aubin and H. Frankowska, Set-valued Analysis,, Birkhäuser Boston, (1990).
|
[5] |
J.-P. Aubin, H. Frankowska and A. Lasota, Poincaré's recurrence theorem for set-valued dynamical systems,, Ann. Polon. Math., 54 (1991), 85.
|
[6] |
L. M. Blumenthal, A new concept in distance geometry with applications to spherical subsets,, Bull. Amer. Math. Soc., 47 (1941), 435.
doi: 10.1090/S0002-9904-1941-07471-9. |
[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. Google Scholar |
[8] |
R. Bowen, Entropy for group endomorphisms and homogeneous spaces,, Trans. Amer. Math. Soc., 153 (1971), 401.
doi: 10.1090/S0002-9947-1971-0274707-X. |
[9] |
M. Brin and G. Stuck, Introduction to Dynamical Systems,, Cambridge University Press, (2002).
doi: 10.1017/CBO9780511755316. |
[10] |
R. S. Burachik and A. N. Iusem, Set-valued Mappings and Enlargements of Monotone Operators,, Springer Optimization and Its Applications, (2008).
|
[11] |
L. J. Cherene, Jr., Set Valued Dynamical Systems and Economic Flow,, Lecture Notes in Economics and Mathematical Systems, (1978).
|
[12] |
M. Ciklová, Dynamical systems generated by functions with connected $G_\delta$ graphs,, Real Anal. Exchange, 30 (): 617.
|
[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, (1867). Google Scholar |
[15] |
E. I. Dinaburg, A correlation between topological entropy and metric entropy (Russian),, Dokl. Akad. Nauk SSSR, 190 (1970), 19.
|
[16] |
T. Downarowicz, Entropy in Dynamical Systems,, New Mathematical Monographs, (2011).
doi: 10.1017/CBO9780511976155. |
[17] |
B. E. Gillam, A new set of postulates for euclidean geometry,, Revista Ci. Lima, 42 (1940), 869.
|
[18] |
A. Katok, Fifty years of entropy in dynamics: 1958-2007,, J. Mod. Dyn., 1 (2007), 545.
doi: 10.3934/jmd.2007.1.545. |
[19] |
A. Katok, Lyapunov exponents, entropy and periodic orbits for diffeomorphisms,, Inst. Hautes Études Sci. Publ. Math., 51 (1980), 137.
|
[20] |
A. Y. Khinchin, On the basic theorems of information theory,, Uspehi Mat. Nauk (N.S.), 11 (1956), 17.
|
[21] |
A. N. Kolmogorov, A new metric invariant of transitive dynamical systems and automorphisms of Lebesgue spaces, (Russian), Topology, 169 (1985), 94.
|
[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.
doi: 10.1137/0314062. |
[23] |
B. McMillan, The basic theorems of information theory,, Ann. Math. Statistics, 24 (1953), 196.
doi: 10.1214/aoms/1177729028. |
[24] |
W. M. Miller, Frobenius-Perron operators and approximation of invariant measures for set-valued dynamical systems,, Set-Valued Anal., 3 (1995), 181.
doi: 10.1007/BF01038599. |
[25] |
W. Miller and E. Akin, Invariant measures for set-valued dynamical systems,, Trans. Amer. Math. Soc., 351 (1999), 1203.
doi: 10.1090/S0002-9947-99-02424-1. |
[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.
|
[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.
|
[28] |
R. T. Rockafellar, Convex Analysis,, Reprint of the 1970 original, (1970).
|
[29] |
Ja. Sinai, On the concept of entropy for a dynamic system, (Russian), Dokl. Akad. Nauk SSSR, 124 (1959), 768.
|
[30] |
Y. Sinai, Kolmogorov-Sinai entropy,, Scholarpedia, 4 (2009). Google Scholar |
[31] |
C. E. Shannon, A mathematical theory of communication,, Bell System Tech. J., 27 (1948), 379.
doi: 10.1002/j.1538-7305.1948.tb01338.x. |
[32] |
E. Tarafdar, P. Watson and X.-Z. Yuan, Poincare's recurrence theorems for set-valued dynamical systems,, Appl. Math. Lett., 10 (1997), 37.
doi: 10.1016/S0893-9659(97)00102-X. |
[33] |
E. Tarafdar and X.-Z. Yuan, The set-valued dynamic system and its applications to Pareto optima,, Acta Appl. Math., 46 (1997), 93.
doi: 10.1023/A:1005722506504. |
[34] |
J. von Neumann, Mathematische Grundlagen der Quantenmechanik,, Unveränderter Nachdruck der ersten Auflage von 1932, (1932).
|
[35] |
P. Walters, An Introduction to Ergodic Theory,, Graduate Texts in Mathematics, (1982).
|
[36] |
A. J. Zaslavski, Convergence of trajectories of discrete dispersive dynamical systems,, Commun. Math. Anal., 4 (2008), 10.
|
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