February  2015, 35(2): 741-755. doi: 10.3934/dcds.2015.35.741

Chain transitive induced interval maps on continua

1. 

University of Toronto, Bahen Centre 40 St. George St., Room 6290, Toronto, ON, M5S 2E4, Canada

2. 

Institute of Mathematics, NASU, Tereshchenkivs'ka 3, 01601, Kyiv, Ukraine

Received  October 2013 Revised  March 2014 Published  September 2014

Let $f:I\rightarrow I$ be a continuous map of a compact interval $I$ and $C(I)$ be the space of all compact subintervals of $I$ with the Hausdorff metric. We investigate chain transitivity of induced maps on subcontinua of $C(I)$. In particular, we prove the following theorem: Let $\mathcal{M}$ be a subcontinuum of $C(I)$ having at most countably many partitioning points. Then, the induced map $\mathcal{F}:C(I)\to C(I)$ $($i.e. $\mathcal{F}(A):=\{f(x):x\in A\}$ for each $A \in C(I)$$)$ is chain transitive on $\mathcal{M}$ iff $\mathcal{F}^{2}\vert_{\mathcal{M}}=Id$.
Citation: Mykola Matviichuk, Damoon Robatian. Chain transitive induced interval maps on continua. Discrete & Continuous Dynamical Systems - A, 2015, 35 (2) : 741-755. doi: 10.3934/dcds.2015.35.741
References:
[1]

G. Acosta, A. Illanes and H. Méndez-Lango, The transitivity of induced maps,, Topology Appl., 156 (2009), 1013. doi: 10.1016/j.topol.2008.12.025. Google Scholar

[2]

E. Akin, Countable metric spaces and chain transitivity,, preprint, (2013). Google Scholar

[3]

S. J. Agronsky, A. M. Bruckner, J. G. Ceder and T. L. Pearson, The structure of $\omega$-limit sets for continuous maps,, Real Analysis Exchange, 15 (): 1989. Google Scholar

[4]

A. D. Barwell, C. Good, P. Oprocha and B. E. Raines, Characterizations of $\omega$-limit sets in topologically hyperbolic systems,, Discrete Contin. Dyn. Syst., 33 (2013), 1819. doi: 10.3934/dcds.2013.33.1819. Google Scholar

[5]

L. S. Block and W. A. Coppel, Dynamics in One Dimension,, Lecture Notes in Mathematics, (1513). Google Scholar

[6]

L. Block and E. Coven, Maps of the interval with every point chain recurrent,, Proc. Amer. Math. Soc., 98 (1986), 513. doi: 10.1090/S0002-9939-1986-0857952-8. Google Scholar

[7]

L. Block and J. Franke, The chain recurrent set, attractors, and explosions,, Ergodic Theory Dynamical Systems, 5 (1985), 321. doi: 10.1017/S0143385700002972. Google Scholar

[8]

A. M. Bruckner and J. Smital, The structure of $\omega$-limit sets for continuous maps of the interval,, Math. Bohemica, 117 (1992), 42. Google Scholar

[9]

Y. N. Dowker and F. G. Friedlander, On limit sets in dynamical systems,, Proc. London Math. Soc. (3), 4 (1954), 168. Google Scholar

[10]

V. V. Fedorenko, Asymptotic periodicity of the trajectories of an interval,, Ukrainian Math. J., 61 (2009), 854. doi: 10.1007/s11253-009-0238-5. Google Scholar

[11]

V. V. Fedorenko, E. Yu. Romanenko and A. N. Sharkovsky, Trajectories of intervals in one-dimensional dynamical systems,, J. Difference Equ. Appl., 13 (2007), 821. doi: 10.1080/10236190701396636. Google Scholar

[12]

A. Illanes and S. B. Nadler, Jr., Hyperspaces. Fundamentals and Recent Advances,, Monographs and Textbooks in Pure and Applied Mathematics, (1999). Google Scholar

[13]

M. Hurley, Chain recurrence and attraction in non-compact spaces,, Ergod. Th. & Dynam. Sys., 11 (1991), 709. doi: 10.1017/S014338570000643X. Google Scholar

[14]

S. Kolyada and D. Robatian, On omega-limit sets of triangular induced maps,, Real Analysis Exchange, 38 (2013), 299. Google Scholar

[15]

S. Kolyada and L. Snoha, On $\omega$-limit sets of triangular maps,, Real Analysis Exchange, 18 (): 1992. Google Scholar

[16]

D. Kwietniak and P. Oprocha, Topological entropy and chaos for maps induced on hyperspaces,, Chaos Solitons Fractals, 33 (2007), 76. doi: 10.1016/j.chaos.2005.12.033. Google Scholar

[17]

M. Matviichuk, On the dynamics of subcontinua of a tree,, J. Difference Equ. Appl., 19 (2013), 223. doi: 10.1080/10236198.2011.634804. Google Scholar

[18]

S. B. Nadler, Jr., Continuum Theory: An Introduction,, Monographs and Textbooks in Pure and Applied Mathematics, (1992). Google Scholar

[19]

A. N. Sharkovsky, Continuous mapping on the limit points of an iteration sequence,, (Russian) Ukrain. Mat. Zh., 18 (1966), 127. Google Scholar

[20]

A. N. Sharkovsky, Partially ordered system of attracting sets,, (Russian) Dokl. Akad. Nauk SSSR, 170 (1966), 1276. Google Scholar

[21]

M. B. Vereikina and A. N. Sharkovsky, The set of almost-recurrent points of a dynamical system,, (Russian. English summary) Dokl. Akad. Nauk Ukrain. SSR Ser. A, 4 (1984), 6. Google Scholar

show all references

References:
[1]

G. Acosta, A. Illanes and H. Méndez-Lango, The transitivity of induced maps,, Topology Appl., 156 (2009), 1013. doi: 10.1016/j.topol.2008.12.025. Google Scholar

[2]

E. Akin, Countable metric spaces and chain transitivity,, preprint, (2013). Google Scholar

[3]

S. J. Agronsky, A. M. Bruckner, J. G. Ceder and T. L. Pearson, The structure of $\omega$-limit sets for continuous maps,, Real Analysis Exchange, 15 (): 1989. Google Scholar

[4]

A. D. Barwell, C. Good, P. Oprocha and B. E. Raines, Characterizations of $\omega$-limit sets in topologically hyperbolic systems,, Discrete Contin. Dyn. Syst., 33 (2013), 1819. doi: 10.3934/dcds.2013.33.1819. Google Scholar

[5]

L. S. Block and W. A. Coppel, Dynamics in One Dimension,, Lecture Notes in Mathematics, (1513). Google Scholar

[6]

L. Block and E. Coven, Maps of the interval with every point chain recurrent,, Proc. Amer. Math. Soc., 98 (1986), 513. doi: 10.1090/S0002-9939-1986-0857952-8. Google Scholar

[7]

L. Block and J. Franke, The chain recurrent set, attractors, and explosions,, Ergodic Theory Dynamical Systems, 5 (1985), 321. doi: 10.1017/S0143385700002972. Google Scholar

[8]

A. M. Bruckner and J. Smital, The structure of $\omega$-limit sets for continuous maps of the interval,, Math. Bohemica, 117 (1992), 42. Google Scholar

[9]

Y. N. Dowker and F. G. Friedlander, On limit sets in dynamical systems,, Proc. London Math. Soc. (3), 4 (1954), 168. Google Scholar

[10]

V. V. Fedorenko, Asymptotic periodicity of the trajectories of an interval,, Ukrainian Math. J., 61 (2009), 854. doi: 10.1007/s11253-009-0238-5. Google Scholar

[11]

V. V. Fedorenko, E. Yu. Romanenko and A. N. Sharkovsky, Trajectories of intervals in one-dimensional dynamical systems,, J. Difference Equ. Appl., 13 (2007), 821. doi: 10.1080/10236190701396636. Google Scholar

[12]

A. Illanes and S. B. Nadler, Jr., Hyperspaces. Fundamentals and Recent Advances,, Monographs and Textbooks in Pure and Applied Mathematics, (1999). Google Scholar

[13]

M. Hurley, Chain recurrence and attraction in non-compact spaces,, Ergod. Th. & Dynam. Sys., 11 (1991), 709. doi: 10.1017/S014338570000643X. Google Scholar

[14]

S. Kolyada and D. Robatian, On omega-limit sets of triangular induced maps,, Real Analysis Exchange, 38 (2013), 299. Google Scholar

[15]

S. Kolyada and L. Snoha, On $\omega$-limit sets of triangular maps,, Real Analysis Exchange, 18 (): 1992. Google Scholar

[16]

D. Kwietniak and P. Oprocha, Topological entropy and chaos for maps induced on hyperspaces,, Chaos Solitons Fractals, 33 (2007), 76. doi: 10.1016/j.chaos.2005.12.033. Google Scholar

[17]

M. Matviichuk, On the dynamics of subcontinua of a tree,, J. Difference Equ. Appl., 19 (2013), 223. doi: 10.1080/10236198.2011.634804. Google Scholar

[18]

S. B. Nadler, Jr., Continuum Theory: An Introduction,, Monographs and Textbooks in Pure and Applied Mathematics, (1992). Google Scholar

[19]

A. N. Sharkovsky, Continuous mapping on the limit points of an iteration sequence,, (Russian) Ukrain. Mat. Zh., 18 (1966), 127. Google Scholar

[20]

A. N. Sharkovsky, Partially ordered system of attracting sets,, (Russian) Dokl. Akad. Nauk SSSR, 170 (1966), 1276. Google Scholar

[21]

M. B. Vereikina and A. N. Sharkovsky, The set of almost-recurrent points of a dynamical system,, (Russian. English summary) Dokl. Akad. Nauk Ukrain. SSR Ser. A, 4 (1984), 6. Google Scholar

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