American Institute of Mathematical Sciences

Advanced
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

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 & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

S. Baldwin, Toward a theory of forcing on maps of trees, Int. J. Bifurcation and Chaos, 8 (1995), 45-56.  Google Scholar

[5]

L. S. Block and W. A. Coppel, Dynamics in One Dimension, Lecture Notes in Math. 1513, Springer-Verlag, Berlin, 1992.  Google Scholar

[6]

S. B. Nadler Jr, Continuum Theory An Introduction, Pure and Appl. Math. 158, Marcel Dekker, New York, 1992.  Google Scholar

show all references

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

S. Baldwin, Toward a theory of forcing on maps of trees, Int. J. Bifurcation and Chaos, 8 (1995), 45-56.  Google Scholar

[5]

L. S. Block and W. A. Coppel, Dynamics in One Dimension, Lecture Notes in Math. 1513, Springer-Verlag, Berlin, 1992.  Google Scholar

[6]

S. B. Nadler Jr, Continuum Theory An Introduction, Pure and Appl. Math. 158, Marcel Dekker, New York, 1992.  Google Scholar

[1]

Vladimír Špitalský. Transitive dendrite map with infinite decomposition ideal. Discrete & Continuous Dynamical Systems, 2015, 35 (2) : 771-792. doi: 10.3934/dcds.2015.35.771
[2]

Denis Gaidashev, Tomas Johnson. Dynamics of the universal area-preserving map associated with period-doubling: Stable sets. Journal of Modern Dynamics, 2009, 3 (4) : 555-587. doi: 10.3934/jmd.2009.3.555
[3]

Andrzej Świȩch. Pointwise properties of $ L^p $-viscosity solutions of uniformly elliptic equations with quadratically growing gradient terms. Discrete & Continuous Dynamical Systems, 2020, 40 (5) : 2945-2962. doi: 10.3934/dcds.2020156
[4]

Alfonso Artigue. Expansive flows of surfaces. Discrete & Continuous Dynamical Systems, 2013, 33 (2) : 505-525. doi: 10.3934/dcds.2013.33.505
[5]

Jorge Groisman. Expansive homeomorphisms of the plane. Discrete & Continuous Dynamical Systems, 2011, 29 (1) : 213-239. doi: 10.3934/dcds.2011.29.213
[6]

Mauricio Achigar. Extensions of expansive dynamical systems. Discrete & Continuous Dynamical Systems, 2021, 41 (7) : 3093-3108. doi: 10.3934/dcds.2020399
[7]

Alfonso Artigue. Lipschitz perturbations of expansive systems. Discrete & Continuous Dynamical Systems, 2015, 35 (5) : 1829-1841. doi: 10.3934/dcds.2015.35.1829
[8]

Amer Rasheed, Aziz Belmiloudi, Fabrice Mahé. Dynamics of dendrite growth in a binary alloy with magnetic field effect. Conference Publications, 2011, 2011 (Special) : 1224-1233. doi: 10.3934/proc.2011.2011.1224
[9]

Haritha C, Nikita Agarwal. Product of expansive Markov maps with hole. Discrete & Continuous Dynamical Systems, 2019, 39 (10) : 5743-5774. doi: 10.3934/dcds.2019252
[10]

Alfonso Artigue. Singular cw-expansive flows. Discrete & Continuous Dynamical Systems, 2017, 37 (6) : 2945-2956. doi: 10.3934/dcds.2017126
[11]

Elon Lindenstrauss. Pointwise theorems for amenable groups. Electronic Research Announcements, 1999, 5: 82-90.

[12]

Ryuichi Suzuki. Universal bounds for quasilinear parabolic equations with convection. Discrete & Continuous Dynamical Systems, 2006, 16 (3) : 563-586. doi: 10.3934/dcds.2006.16.563
[13]

Hai-Liang Wu, Zhi-Wei Sun. Some universal quadratic sums over the integers. Electronic Research Archive, 2019, 27: 69-87. doi: 10.3934/era.2019010
[14]

S. Eigen, A. B. Hajian, V. S. Prasad. Universal skyscraper templates for infinite measure preserving transformations. Discrete & Continuous Dynamical Systems, 2006, 16 (2) : 343-360. doi: 10.3934/dcds.2006.16.343
[15]

Thierry Cazenave, Flávio Dickstein, Fred B. Weissler. Universal solutions of the heat equation on $\mathbb R^N$. Discrete & Continuous Dynamical Systems, 2003, 9 (5) : 1105-1132. doi: 10.3934/dcds.2003.9.1105
[16]

Nusret Balci, Ciprian Foias, M. S Jolly, Ricardo Rosa. On universal relations in 2-D turbulence. Discrete & Continuous Dynamical Systems, 2010, 27 (4) : 1327-1351. doi: 10.3934/dcds.2010.27.1327
[17]

Jorge Groisman. Expansive and fixed point free homeomorphisms of the plane. Discrete & Continuous Dynamical Systems, 2012, 32 (5) : 1709-1721. doi: 10.3934/dcds.2012.32.1709
[18]

Alfonso Artigue. Anomalous cw-expansive surface homeomorphisms. Discrete & Continuous Dynamical Systems, 2016, 36 (7) : 3511-3518. doi: 10.3934/dcds.2016.36.3511
[19]

Woochul Jung, Ngocthach Nguyen, Yinong Yang. Spectral decomposition for rescaling expansive flows with rescaled shadowing. Discrete & Continuous Dynamical Systems, 2020, 40 (4) : 2267-2283. doi: 10.3934/dcds.2020113
[20]

Alfonso Artigue. Robustly N-expansive surface diffeomorphisms. Discrete & Continuous Dynamical Systems, 2016, 36 (5) : 2367-2376. doi: 10.3934/dcds.2016.36.2367

2019 Impact Factor: 1.338

Tools

Metrics

  • PDF downloads (56)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences