The structure of dendrites constructed by pointwise P-expansive maps on the unit interval

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January 2016, 36(1): 43-61. doi: 10.3934/dcds.2016.36.43

The structure of dendrites constructed by pointwise P-expansive maps on the unit interval

Tatsuya Arai ^1,

1.	Research and Support Center on Higher Education, for the Hearing Impaired and Visually Impaired, Tsukuba University of Technology, Ibaraki 305-8520, Japan

Received May 2014 Revised March 2015 Published June 2015

Abstract
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Let $f$ be a continuous map from the unit interval to itself. In this paper, we investigate the structure of space $Z$ which is constructed corresponding to the behaviors of $f$ and a periodic orbit $P$ of $f$. Under some restriction of $f$, we get necessary and sufficient conditions for $Z$ being the universal dendrite. Furthermore $Z$ is classified into five types especially when it is a tree.

Keywords: universal dendrite., Pointwise $P$-expansive map.

Mathematics Subject Classification: Primary: 37E05, 54C05; Secondary: 54F5.

Citation: Tatsuya Arai. The structure of dendrites constructed by pointwise P-expansive maps on the unit interval. Discrete and Continuous Dynamical Systems, 2016, 36 (1) : 43-61. doi: 10.3934/dcds.2016.36.43

References:

[1]	L. Alseda, 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]	T. Arai and N. Chinen, The construction of chaotic maps in the sense of Devaney on dendrites which commute to continuous maps on the unit interval, Discrete Continuous Dynam. Systems - A, 11 (2004), 547-556. doi: 10.3934/dcds.2004.11.547.
[3]	T. Arai, N. Chinen, H. Kato and K. Yokoi, The construction of P-expansive maps of regular continua : A geometric approach, Topology Appl., 103 (2000), 309-321. doi: 10.1016/S0166-8641(99)00004-8.
[4]	S. Baldwin, Toward a theory of forcing on maps of trees, Int. J. Bifurcation and Chaos, 8 (1995), 45-56.
[5]	L. S. Block and W. A. Coppel, Dynamics in One Dimension, Lecture Notes in Math. 1513, Springer-Verlag, Berlin, 1992.
[6]	S. B. Nadler Jr, Continuum Theory An Introduction, Pure and Appl. Math. 158, Marcel Dekker, New York, 1992.

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References:

[1]	L. Alseda, 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]	T. Arai and N. Chinen, The construction of chaotic maps in the sense of Devaney on dendrites which commute to continuous maps on the unit interval, Discrete Continuous Dynam. Systems - A, 11 (2004), 547-556. doi: 10.3934/dcds.2004.11.547.
[3]	T. Arai, N. Chinen, H. Kato and K. Yokoi, The construction of P-expansive maps of regular continua : A geometric approach, Topology Appl., 103 (2000), 309-321. doi: 10.1016/S0166-8641(99)00004-8.
[4]	S. Baldwin, Toward a theory of forcing on maps of trees, Int. J. Bifurcation and Chaos, 8 (1995), 45-56.
[5]	L. S. Block and W. A. Coppel, Dynamics in One Dimension, Lecture Notes in Math. 1513, Springer-Verlag, Berlin, 1992.
[6]	S. B. Nadler Jr, Continuum Theory An Introduction, Pure and Appl. Math. 158, Marcel Dekker, New York, 1992.

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