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On the stability of periodic orbits in delay equations with large delay
Entropy and exact Devaney chaos on totally regular continua
1. | Department of Mathematics, Faculty of Natural Sciences, Matej Bel University, Tajovského 40, 974 01 Banská Bystrica |
References:
[1] |
L. Alsedà, S. Baldwin, J. Llibre and M. Misiurewicz, Entropy of transitive tree maps, Topology, 36 (1997), 519-532.
doi: 10.1016/0040-9383(95)00070-4. |
[2] |
L. Alsedà, S. Kolyada, J. Llibre and L'. Snoha, Entropy and periodic points for transitive maps, Trans. Amer. Math. Soc., 351 (1999), 1551-1573.
doi: 10.1090/S0002-9947-99-02077-2. |
[3] |
Ll. Alsedà, J. Llibre and M. Misiurewicz, "Combinatorial Dynamics and Entropy in Dimension One," $2^{nd}$ edition, Advanced Series in Nonlinear Dynamics 5, World Scientific, Singapore, 2000. |
[4] |
L. Alsedà, M. A. del Río and J. A. Rodríguez, A splitting theorem for transitive maps, J. Math. Anal. Appl., 232 (1999), 359-375.
doi: 10.1006/jmaa.1999.6277. |
[5] |
S. Baldwin, Entropy estimates for transitive maps on trees, Topology, 40 (2001), 551-569.
doi: 10.1016/S0040-9383(99)00074-9. |
[6] |
F. Balibrea and L'. Snoha, Topological entropy of Devaney chaotic maps, Topology Appl., 133 (2003), 225-239.
doi: 10.1016/S0166-8641(03)00090-7. |
[7] |
R. H. Bing, Partitioning a set, Bull. Amer. Math. Soc., 55 (1949), 1101-1110. |
[8] |
A. Blokh, On sensitive mappings of the interval, Russian Math. Surveys, 37 (1982), 203-204. |
[9] |
A. Blokh, On transitive mappings of one-dimensional branched manifolds, (Russian), Differential-Difference Equations and Problems of Mathematical Physics, 3-9, 131, Akad. Nauk Ukrain. SSR, Inst. Mat., Kiev, (1984). |
[10] |
A. Blokh, On the connection between entropy and transitivity for one-dimensional mappings, Russ. Math. Surv., 42 (1987), 165-166. |
[11] |
R. Bowen, Entropy for group endomorphisms and homogeneous spaces, Trans. Amer. Math. Soc., 153 (1971), 401-414; Erratum: Trans. Amer. Math. Soc. 181 (1973), 509-510. |
[12] |
R. D. Buskirk, J. Nikiel and E. D. Tymchatyn, Totally regular curves as inverse limits, Houston J. Math., 18 (1992), 319-327. |
[13] |
E. I. Dinaburg, A connection between various entropy characterizations of dynamical systems, (Russian), Izv. Akad. Nauk SSSR Ser. Mat., 35 (1971), 324-366. |
[14] |
M. Dirbák, L'. Snoha and V. Špitalský, Minimality, transitivity, mixing and topological entropy on spaces with a free interval,, Ergodic Theory Dynam. Systems, ().
doi: 10.1017/S0143385712000442. |
[15] |
K. J. Falconer, "The Geometry of Fractal Sets," Cambridge University Press, Cambridge, 1986. |
[16] |
H. Federer, "Geometric Measure Theory," Springer-Verlag New York Inc., New York, 1969. |
[17] |
D. H. Fremlin, Spaces of finite length, Proc. London Math. Soc., 64 (1992), 449-486.
doi: 10.1112/plms/s3-64.3.449. |
[18] |
G. Harańczyk, D. Kwietniak and P. Oprocha, Topological structure and entropy of mixing graph maps,, preprint, ().
|
[19] |
A. Katok and B. Hasselblatt, "Introduction to the Modern Theory of Dynamical Systems," Cambridge University Press, Cambridge, 1995. |
[20] |
S. Kolyada and M. Matviichuk, On extensions of transitive maps, Discrete Contin. Dyn. Syst., 30 (2011), 767-777.
doi: 10.3934/dcds.2011.30.767. |
[21] |
K. Kuratowski, "Topology, vol. 2," Academic Press and PWN, Warszawa, 1968. |
[22] |
D. Kwietniak and M. Misiurewicz, Exact Devaney chaos and entropy, Qual. Theory Dyn. Syst., 6 (2005), 169-179.
doi: 10.1007/BF02972670. |
[23] |
S. Macías, "Topics on Continua," Chapman & Hall/CRC, Boca Raton, FL, 2005.
doi: 10.1201/9781420026535. |
[24] |
P. Mattila, "Geometry of Sets and Measures in Euclidean Spaces. Fractals and Rectifiability," Cambridge University Press, Cambridge, 1995. |
[25] |
S. B. Nadler, "Continuum Theory. An Introduction," Monographs and Textbooks in Pure and Applied Mathematics, 158, Marcel Dekker, Inc., New York, 1992. |
[26] |
J. Nikiel, Locally connected curves viewed as inverse limits, Fund. Math., 133 (1989), 125-134. |
[27] |
S. Ruette, Chaos for continuous interval maps - a survey of relationship between the various sorts of chaos,, preprint, ().
|
[28] |
V. Špitalský, Length-expanding Lipschitz maps on totally regular continua,, preprint, ().
|
[29] |
G. T. Whyburn, "Analytic Topology," American Mathematical Society, New York, 1942. |
[30] |
X. Ye, Topological entropy of transitive maps of a tree, Ergodic Theory Dynam. Systems, 20 (2000), 289-314.
doi: 10.1017/S0143385700000134. |
show all references
References:
[1] |
L. Alsedà, S. Baldwin, J. Llibre and M. Misiurewicz, Entropy of transitive tree maps, Topology, 36 (1997), 519-532.
doi: 10.1016/0040-9383(95)00070-4. |
[2] |
L. Alsedà, S. Kolyada, J. Llibre and L'. Snoha, Entropy and periodic points for transitive maps, Trans. Amer. Math. Soc., 351 (1999), 1551-1573.
doi: 10.1090/S0002-9947-99-02077-2. |
[3] |
Ll. Alsedà, J. Llibre and M. Misiurewicz, "Combinatorial Dynamics and Entropy in Dimension One," $2^{nd}$ edition, Advanced Series in Nonlinear Dynamics 5, World Scientific, Singapore, 2000. |
[4] |
L. Alsedà, M. A. del Río and J. A. Rodríguez, A splitting theorem for transitive maps, J. Math. Anal. Appl., 232 (1999), 359-375.
doi: 10.1006/jmaa.1999.6277. |
[5] |
S. Baldwin, Entropy estimates for transitive maps on trees, Topology, 40 (2001), 551-569.
doi: 10.1016/S0040-9383(99)00074-9. |
[6] |
F. Balibrea and L'. Snoha, Topological entropy of Devaney chaotic maps, Topology Appl., 133 (2003), 225-239.
doi: 10.1016/S0166-8641(03)00090-7. |
[7] |
R. H. Bing, Partitioning a set, Bull. Amer. Math. Soc., 55 (1949), 1101-1110. |
[8] |
A. Blokh, On sensitive mappings of the interval, Russian Math. Surveys, 37 (1982), 203-204. |
[9] |
A. Blokh, On transitive mappings of one-dimensional branched manifolds, (Russian), Differential-Difference Equations and Problems of Mathematical Physics, 3-9, 131, Akad. Nauk Ukrain. SSR, Inst. Mat., Kiev, (1984). |
[10] |
A. Blokh, On the connection between entropy and transitivity for one-dimensional mappings, Russ. Math. Surv., 42 (1987), 165-166. |
[11] |
R. Bowen, Entropy for group endomorphisms and homogeneous spaces, Trans. Amer. Math. Soc., 153 (1971), 401-414; Erratum: Trans. Amer. Math. Soc. 181 (1973), 509-510. |
[12] |
R. D. Buskirk, J. Nikiel and E. D. Tymchatyn, Totally regular curves as inverse limits, Houston J. Math., 18 (1992), 319-327. |
[13] |
E. I. Dinaburg, A connection between various entropy characterizations of dynamical systems, (Russian), Izv. Akad. Nauk SSSR Ser. Mat., 35 (1971), 324-366. |
[14] |
M. Dirbák, L'. Snoha and V. Špitalský, Minimality, transitivity, mixing and topological entropy on spaces with a free interval,, Ergodic Theory Dynam. Systems, ().
doi: 10.1017/S0143385712000442. |
[15] |
K. J. Falconer, "The Geometry of Fractal Sets," Cambridge University Press, Cambridge, 1986. |
[16] |
H. Federer, "Geometric Measure Theory," Springer-Verlag New York Inc., New York, 1969. |
[17] |
D. H. Fremlin, Spaces of finite length, Proc. London Math. Soc., 64 (1992), 449-486.
doi: 10.1112/plms/s3-64.3.449. |
[18] |
G. Harańczyk, D. Kwietniak and P. Oprocha, Topological structure and entropy of mixing graph maps,, preprint, ().
|
[19] |
A. Katok and B. Hasselblatt, "Introduction to the Modern Theory of Dynamical Systems," Cambridge University Press, Cambridge, 1995. |
[20] |
S. Kolyada and M. Matviichuk, On extensions of transitive maps, Discrete Contin. Dyn. Syst., 30 (2011), 767-777.
doi: 10.3934/dcds.2011.30.767. |
[21] |
K. Kuratowski, "Topology, vol. 2," Academic Press and PWN, Warszawa, 1968. |
[22] |
D. Kwietniak and M. Misiurewicz, Exact Devaney chaos and entropy, Qual. Theory Dyn. Syst., 6 (2005), 169-179.
doi: 10.1007/BF02972670. |
[23] |
S. Macías, "Topics on Continua," Chapman & Hall/CRC, Boca Raton, FL, 2005.
doi: 10.1201/9781420026535. |
[24] |
P. Mattila, "Geometry of Sets and Measures in Euclidean Spaces. Fractals and Rectifiability," Cambridge University Press, Cambridge, 1995. |
[25] |
S. B. Nadler, "Continuum Theory. An Introduction," Monographs and Textbooks in Pure and Applied Mathematics, 158, Marcel Dekker, Inc., New York, 1992. |
[26] |
J. Nikiel, Locally connected curves viewed as inverse limits, Fund. Math., 133 (1989), 125-134. |
[27] |
S. Ruette, Chaos for continuous interval maps - a survey of relationship between the various sorts of chaos,, preprint, ().
|
[28] |
V. Špitalský, Length-expanding Lipschitz maps on totally regular continua,, preprint, ().
|
[29] |
G. T. Whyburn, "Analytic Topology," American Mathematical Society, New York, 1942. |
[30] |
X. Ye, Topological entropy of transitive maps of a tree, Ergodic Theory Dynam. Systems, 20 (2000), 289-314.
doi: 10.1017/S0143385700000134. |
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