Ogasawara, Yoshihito

Oishi, Shin'ichi

Kyoto University Research Information Repository

Title On a characteristic property of the tent map (General andGeometric Topology today and their problems)Author(s)Ogasawara, Yoshihito; Oishi, Shin'ichiCitation数理解析研究所講究録 (2013), 1833: 98-103Issue Date2013-05URL http://hdl.handle.net/2433/194862RightType Departmental Bulletin PaperTextversionpublisherKyoto UniversityOn a characteristic property of the tent mapYoshihito Ogasawara and Shin’ichi OishiFaculty of Science and EngineeringWaseda UniversityAs a generalization of a characteristic property of fundamental chaotic maps suchas the tent map, a primitive chaos is defined by $(X, \{X_{\lambda}, \lambda\in\Lambda\}, \{f_{X_{\lambda}}, \lambda\in\Lambda\}))$which satisfies the following property, where $X$ is a set, $\{X_{\lambda}, \lambda\in\Lambda\}$ is a family ofsubsets of $X$ which contains no empty set, and $\{f_{X_{\lambda}}, \lambda\in\Lambda\}$ is a family of maps$f_{X_{\lambda}}:X_{\lambda}arrow X[1].$(P) For any infinite sequence $\omega_{0},$ $\omega_{1},$ $\omega_{2},$ $\ldots$ of sets, there exists an initial point$x_{0}\in\omega_{0}$ such that $f_{\omega_{0}}(x_{0})\in\omega_{1},$ $f_{\omega_{1}}(f_{\omega_{0}}(x_{0}))\in\omega_{2},$ $\ldots$ , where $\omega_{i}\in\{X_{\lambda}, \lambda\in\Lambda\}$for each $i.$The primitive chaos is closely related to general problems about determinism, causal-ity, free will, and irreversibility [1], which are vital problems in science (e.g., refs.[2-26] $)$ .Then, the following propositions present sufficient conditions for guaranteeing theprimitive chaos.Proposition 1 [1]. If $X$ is a nondegenemte Peano continuum, for any $\epsilon>0$ , thereexist finitely many nondegenerate Peano subcontinua $X_{1},$ $\ldots,$ $X_{n}$ covering $X$ suchthat $diaX_{i}<\epsilon,$ $i=1,$ $\ldots,$ $n$ . Then, for each $i$ , for any positive integer $n^{i}$ , for any$n^{i}$ points $x_{1}^{i},$ $\ldots,$ $x_{n^{i}}^{i}\in X_{i}$ and $y_{1}^{i},$ $\ldots,$ $y_{n^{i}}^{i}\in X$ , there exists a continuous surjection$f_{X_{i}}$ : $X_{i}arrow X$ such that $f_{X_{i}}(x_{1}^{i})=y_{1}^{i},$ $\ldots,$ $f_{X_{i}}(x_{n^{i}}^{i})=y_{n^{i}}^{i}$ , and they satisfy theproperty (P).1833 2013 98-103 98Proposition 2 [27]. If $X$ is a Cantor set, for any positive integer $n$ , there exist apartition $\{X_{1}, \ldots, X_{n}\}$ of $X$ and maps $f_{X_{i}}$ : $X_{i}arrow X,$ $i=1,$ $\ldots,$ $n$ , and they satisfythe property $(P)$ .Here, a nondegenerate space means the space that consists of more than one point,a Peano continuum is a locally connected continuum, and a continuum is a nonemptyconnected compact metric space. A Cantor set is any space that is homeomorphicto the Cantor middle-third set, and a space is a Cantor set if and only if it is azero-dimensional, perfect compact metric space [28, Theorem 8.1].Proposition 1 explains the reason why we are surrounded by diverse chaotic be-haviors [29], and Proposition 2 implies the possibility of the Cantor set for a newrecognition of natural phenomena.AcknowlegementThis study was supported by Japan Science and Technology Agency.References[1] Y. Ogasawara: J. Phys. Soc. Jpn. 79 (2010) 15002.[2] P. S. 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Heider): Principles of TopologicalPsychology (McGraw-Hill Book Company, New York, 1936).[31] J. Piaget: Le Structumlisme (Presses Universitaires de Rance, Paris, 1968) [inFrench].101[32] R. Thom: Stabilit\’e Structurelle et Morphog\’en\‘ese (InterEditions, Paris, 1977)[in French].[33] S. B. Nadler, Jr.: Continuum Theorlt (Marcel Dekker Inc., New York, 1992).[34] Y. Ogasawara and S. Oishi: J. Phys. Soc. Jpn. 80 (2011) 67002.[35] Y. Ogasawara: Mono no Mikata Toshiteno Isokukanron Nyumon (Baifukan,Tokyo, 2011) [in Japanese].102Faculty of Science and EngineeringWaseda UniversityOhkubo, Shinjuku-ku, Tokyo 169-8555 JapanE–mail address: ogasawara@aoni.waseda.jp$E\ovalbox{\tt\small REJECT}\tilde{I}^{r}6E\exists*\not\cong\cdot\iota\ovalbox{\tt\small REJECT} Ig_{f/}’\lceil\check{|y}\beta_{\pi}^{g}$ $/\rfloor)\ovalbox{\tt\small REJECT}_{/}E_{\backslash }\ovalbox{\tt\small REJECT}(_{-}^{-},$ $*E^{\backslash }\not\in-$103

On a characteristic property of the tent map (General and Geometric Topology today and their problems)

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On a characteristic property of the tent map (General and Geometric Topology today and their problems)

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