# American Institute of Mathematical Sciences

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. [2] E. Akin, Countable metric spaces and chain transitivity,, preprint, (2013). [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. [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. [5] L. S. Block and W. A. Coppel, Dynamics in One Dimension,, Lecture Notes in Mathematics, (1513). [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. [7] L. Block and J. Franke, The chain recurrent set, attractors, and explosions,, Ergodic Theory Dynamical Systems, 5 (1985), 321. doi: 10.1017/S0143385700002972. [8] A. M. Bruckner and J. Smital, The structure of $\omega$-limit sets for continuous maps of the interval,, Math. Bohemica, 117 (1992), 42. [9] Y. N. Dowker and F. G. Friedlander, On limit sets in dynamical systems,, Proc. London Math. Soc. (3), 4 (1954), 168. [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. [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. [12] A. Illanes and S. B. Nadler, Jr., Hyperspaces. Fundamentals and Recent Advances,, Monographs and Textbooks in Pure and Applied Mathematics, (1999). [13] M. Hurley, Chain recurrence and attraction in non-compact spaces,, Ergod. Th. & Dynam. Sys., 11 (1991), 709. doi: 10.1017/S014338570000643X. [14] S. Kolyada and D. Robatian, On omega-limit sets of triangular induced maps,, Real Analysis Exchange, 38 (2013), 299. [15] S. Kolyada and L. Snoha, On $\omega$-limit sets of triangular maps,, Real Analysis Exchange, 18 (): 1992. [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. [17] M. Matviichuk, On the dynamics of subcontinua of a tree,, J. Difference Equ. Appl., 19 (2013), 223. doi: 10.1080/10236198.2011.634804. [18] S. B. Nadler, Jr., Continuum Theory: An Introduction,, Monographs and Textbooks in Pure and Applied Mathematics, (1992). [19] A. N. Sharkovsky, Continuous mapping on the limit points of an iteration sequence,, (Russian) Ukrain. Mat. Zh., 18 (1966), 127. [20] A. N. Sharkovsky, Partially ordered system of attracting sets,, (Russian) Dokl. Akad. Nauk SSSR, 170 (1966), 1276. [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.

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##### 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. [2] E. Akin, Countable metric spaces and chain transitivity,, preprint, (2013). [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. [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. [5] L. S. Block and W. A. Coppel, Dynamics in One Dimension,, Lecture Notes in Mathematics, (1513). [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. [7] L. Block and J. Franke, The chain recurrent set, attractors, and explosions,, Ergodic Theory Dynamical Systems, 5 (1985), 321. doi: 10.1017/S0143385700002972. [8] A. M. Bruckner and J. Smital, The structure of $\omega$-limit sets for continuous maps of the interval,, Math. Bohemica, 117 (1992), 42. [9] Y. N. Dowker and F. G. Friedlander, On limit sets in dynamical systems,, Proc. London Math. Soc. (3), 4 (1954), 168. [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. [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. [12] A. Illanes and S. B. Nadler, Jr., Hyperspaces. Fundamentals and Recent Advances,, Monographs and Textbooks in Pure and Applied Mathematics, (1999). [13] M. Hurley, Chain recurrence and attraction in non-compact spaces,, Ergod. Th. & Dynam. Sys., 11 (1991), 709. doi: 10.1017/S014338570000643X. [14] S. Kolyada and D. Robatian, On omega-limit sets of triangular induced maps,, Real Analysis Exchange, 38 (2013), 299. [15] S. Kolyada and L. Snoha, On $\omega$-limit sets of triangular maps,, Real Analysis Exchange, 18 (): 1992. [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. [17] M. Matviichuk, On the dynamics of subcontinua of a tree,, J. Difference Equ. Appl., 19 (2013), 223. doi: 10.1080/10236198.2011.634804. [18] S. B. Nadler, Jr., Continuum Theory: An Introduction,, Monographs and Textbooks in Pure and Applied Mathematics, (1992). [19] A. N. Sharkovsky, Continuous mapping on the limit points of an iteration sequence,, (Russian) Ukrain. Mat. Zh., 18 (1966), 127. [20] A. N. Sharkovsky, Partially ordered system of attracting sets,, (Russian) Dokl. Akad. Nauk SSSR, 170 (1966), 1276. [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.
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