On a dendrite generated by a zero-dimensional weak self-similar set

Akihiko Kitada; Yoshihito Ogasawara; Tomoyuki Yamamoto

doi:10.1016/j.chaos.2006.05.029

On a dendrite generated by a zero-dimensional weak self-similar set

Akihiko Kitada^*, Yoshihito Ogasawara, Tomoyuki Yamamoto

^*Corresponding author for this work

School of Fundamental Science and Engineering

Research output: Contribution to journal › Article › peer-review

4 Citations (Scopus)

Abstract

Let S be a zero-dimensional, perfect, compact weak self-similar set generated in dendrite X by a family {f_j} of weak contractions from X to itself. Decomposition space D_f of S due to a continuous mapping f from S onto X is also a dendrite. In the dendrite D_f, there exists a zero-dimensional, perfect, compact weak self-similar set S¹ based on a family {f_j¹} each of which is topologically conjugate to f_j. Decomposition space D_f¹ of S¹ due to a continuous mapping f¹ from S¹ onto D_f is again a dendrite. In this manner, through the successive formation of weak self-similar set, we can obtain a sequence X, D_f, D_f¹, ... of dendrite any pair in which are mutually homeomorphic.

Original language	English
Pages (from-to)	1732-1735
Number of pages	4
Journal	Chaos, solitons and fractals
Volume	34
Issue number	5
DOIs	https://doi.org/10.1016/j.chaos.2006.05.029
Publication status	Published - 2007 Dec

ASJC Scopus subject areas

Statistical and Nonlinear Physics
Mathematics(all)
Physics and Astronomy(all)
Applied Mathematics

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10.1016/j.chaos.2006.05.029

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abstract = "Let S be a zero-dimensional, perfect, compact weak self-similar set generated in dendrite X by a family {fj} of weak contractions from X to itself. Decomposition space Df of S due to a continuous mapping f from S onto X is also a dendrite. In the dendrite Df, there exists a zero-dimensional, perfect, compact weak self-similar set S1 based on a family {fj1} each of which is topologically conjugate to fj. Decomposition space Df1 of S1 due to a continuous mapping f1 from S1 onto Df is again a dendrite. In this manner, through the successive formation of weak self-similar set, we can obtain a sequence X, Df, Df1, ... of dendrite any pair in which are mutually homeomorphic.",

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AU - Yamamoto, Tomoyuki

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N2 - Let S be a zero-dimensional, perfect, compact weak self-similar set generated in dendrite X by a family {fj} of weak contractions from X to itself. Decomposition space Df of S due to a continuous mapping f from S onto X is also a dendrite. In the dendrite Df, there exists a zero-dimensional, perfect, compact weak self-similar set S1 based on a family {fj1} each of which is topologically conjugate to fj. Decomposition space Df1 of S1 due to a continuous mapping f1 from S1 onto Df is again a dendrite. In this manner, through the successive formation of weak self-similar set, we can obtain a sequence X, Df, Df1, ... of dendrite any pair in which are mutually homeomorphic.

AB - Let S be a zero-dimensional, perfect, compact weak self-similar set generated in dendrite X by a family {fj} of weak contractions from X to itself. Decomposition space Df of S due to a continuous mapping f from S onto X is also a dendrite. In the dendrite Df, there exists a zero-dimensional, perfect, compact weak self-similar set S1 based on a family {fj1} each of which is topologically conjugate to fj. Decomposition space Df1 of S1 due to a continuous mapping f1 from S1 onto Df is again a dendrite. In this manner, through the successive formation of weak self-similar set, we can obtain a sequence X, Df, Df1, ... of dendrite any pair in which are mutually homeomorphic.

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On a dendrite generated by a zero-dimensional weak self-similar set

Abstract

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