Chain transitive induced interval maps on continua

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

Chain transitive induced interval maps on continua

Mykola Matviichuk ^1, and Damoon Robatian ^2,

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

Abstract
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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 $\mathcal{F}^{2}\vert_{\mathcal{M}}=Id$.

Keywords: chain transitivity, continuum., Dynamics of sets, induced maps.

Mathematics Subject Classification: Primary: 37B20; Secondary: 37B4.

Citation: Mykola Matviichuk, Damoon Robatian. Chain transitive induced interval maps on continua. Discrete and Continuous Dynamical Systems, 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), 1013-1033. 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), 1819-1833. doi: 10.3934/dcds.2013.33.1819.
[5]	L. S. Block and W. A. Coppel, Dynamics in One Dimension, Lecture Notes in Mathematics, Vol. 1513, Berlin Heidelberg, Springer-Verlag, 1992.
[6]	L. Block and E. Coven, Maps of the interval with every point chain recurrent, Proc. Amer. Math. Soc., 98 (1986), 513-515. 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), 321-327. 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), 42-47.
[9]	Y. N. Dowker and F. G. Friedlander, On limit sets in dynamical systems, Proc. London Math. Soc. (3), 4 (1954), 168-176.
[10]	V. V. Fedorenko, Asymptotic periodicity of the trajectories of an interval, Ukrainian Math. J., 61 (2009), 854-858. 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), 821-828. 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, 216, Marcel Dekker, Inc., New York, 1999.
[13]	M. Hurley, Chain recurrence and attraction in non-compact spaces, Ergod. Th. & Dynam. Sys., 11 (1991), 709-729. doi: 10.1017/S014338570000643X.
[14]	S. Kolyada and D. Robatian, On omega-limit sets of triangular induced maps, Real Analysis Exchange, 38 (2013), 299-316.
[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), 76-86. 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), 223-233. doi: 10.1080/10236198.2011.634804.
[18]	S. B. Nadler, Jr., Continuum Theory: An Introduction, Monographs and Textbooks in Pure and Applied Mathematics, 158, Marcel Dekker, New York, 1992.
[19]	A. N. Sharkovsky, Continuous mapping on the limit points of an iteration sequence, (Russian) Ukrain. Mat. Zh., 18 (1966), 127-130.
[20]	A. N. Sharkovsky, Partially ordered system of attracting sets, (Russian) Dokl. Akad. Nauk SSSR, 170 (1966), 1276-1278.
[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), 6-9.

show all references

References:

[1]	G. Acosta, A. Illanes and H. Méndez-Lango, The transitivity of induced maps, Topology Appl., 156 (2009), 1013-1033. 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), 1819-1833. doi: 10.3934/dcds.2013.33.1819.
[5]	L. S. Block and W. A. Coppel, Dynamics in One Dimension, Lecture Notes in Mathematics, Vol. 1513, Berlin Heidelberg, Springer-Verlag, 1992.
[6]	L. Block and E. Coven, Maps of the interval with every point chain recurrent, Proc. Amer. Math. Soc., 98 (1986), 513-515. 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), 321-327. 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), 42-47.
[9]	Y. N. Dowker and F. G. Friedlander, On limit sets in dynamical systems, Proc. London Math. Soc. (3), 4 (1954), 168-176.
[10]	V. V. Fedorenko, Asymptotic periodicity of the trajectories of an interval, Ukrainian Math. J., 61 (2009), 854-858. 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), 821-828. 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, 216, Marcel Dekker, Inc., New York, 1999.
[13]	M. Hurley, Chain recurrence and attraction in non-compact spaces, Ergod. Th. & Dynam. Sys., 11 (1991), 709-729. doi: 10.1017/S014338570000643X.
[14]	S. Kolyada and D. Robatian, On omega-limit sets of triangular induced maps, Real Analysis Exchange, 38 (2013), 299-316.
[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), 76-86. 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), 223-233. doi: 10.1080/10236198.2011.634804.
[18]	S. B. Nadler, Jr., Continuum Theory: An Introduction, Monographs and Textbooks in Pure and Applied Mathematics, 158, Marcel Dekker, New York, 1992.
[19]	A. N. Sharkovsky, Continuous mapping on the limit points of an iteration sequence, (Russian) Ukrain. Mat. Zh., 18 (1966), 127-130.
[20]	A. N. Sharkovsky, Partially ordered system of attracting sets, (Russian) Dokl. Akad. Nauk SSSR, 170 (1966), 1276-1278.
[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), 6-9.

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