Artículo Original
Publicado 2003-10-10
Palabras clave
- entire functions,
- chaotic maps
Cómo citar
Méndez-Lango, H. (2003). On Intervals, Sensitivity Implies Chaos. Revista Integración, Temas De matemáticas, 21(1 y 2), 15–23. Recuperado a partir de https://revistas.uis.edu.co/index.php/revistaintegracion/article/view/530
Resumen
In this note we investigate which properties can be derived for a continuous function f defined on an interval I if the only a priori given information is its sensitive dependence on initial conditions. Our main result is the following: If f is sensitive, then f is chaotic, in the sense of Devaney, on a nonempty interior subset of I; the set of aperiodic points is dense in I as well as the set of asintotically periodic points; moreover, f has positive topological entropy.
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Referencias
[1]Banks J., Brooks J., Cairns G., Davis G.andStacey P.“On Devaney’sDefinition of Chaos”,Amer. Math. Monthly, 1992, 332–334.
[2]Block L. S.andCoppel W. A.Dynamics in One Dimension, Lecture Notesin Math. 1523, Springer Verlag, 1991.
[3]Devaney R. L.An Introduction to Chaotic Dynamical Systems, AddisonWesley, 1989.[4]Hocking J. G.andYoung G. S.Topology,Dover Publications, Inc., NewYork, 1988.
[5]Méndez-Lango H.“Las Quebraditas (propiedades dinámicas de una pecu-liar familia de funciones en el intervalo)”,Miscelánea Matemática,35(2002),59–71.
[6]Misiurewicz M.“Invariant Measures for Continuous Transformations of[0,1]with Zero Topological Entropy”,Lecture Notes in Math.,729(1980), 144–152.
[7]Nitecki Z.Topological Dynamics on the Interval, Ergodic Theory andDynamical Systems II, Proc. Special Year, Maryland, 1979-1980 (A. Katoked.) Birkhäuser, Basel, 1982, 1–73.
[8]Vellekook M.andBerglund R.“On Intervals, Transitivity=Chaos”,Amer. Math. Monthly,101(1994), 353–355.
[9]Walters P.,An Introduction to Ergodic Theory, Graduate Texts in Math.,79, Springer Verlag, 1982.
[2]Block L. S.andCoppel W. A.Dynamics in One Dimension, Lecture Notesin Math. 1523, Springer Verlag, 1991.
[3]Devaney R. L.An Introduction to Chaotic Dynamical Systems, AddisonWesley, 1989.[4]Hocking J. G.andYoung G. S.Topology,Dover Publications, Inc., NewYork, 1988.
[5]Méndez-Lango H.“Las Quebraditas (propiedades dinámicas de una pecu-liar familia de funciones en el intervalo)”,Miscelánea Matemática,35(2002),59–71.
[6]Misiurewicz M.“Invariant Measures for Continuous Transformations of[0,1]with Zero Topological Entropy”,Lecture Notes in Math.,729(1980), 144–152.
[7]Nitecki Z.Topological Dynamics on the Interval, Ergodic Theory andDynamical Systems II, Proc. Special Year, Maryland, 1979-1980 (A. Katoked.) Birkhäuser, Basel, 1982, 1–73.
[8]Vellekook M.andBerglund R.“On Intervals, Transitivity=Chaos”,Amer. Math. Monthly,101(1994), 353–355.
[9]Walters P.,An Introduction to Ergodic Theory, Graduate Texts in Math.,79, Springer Verlag, 1982.