American Institute of Mathematical Sciences

Advanced
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

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 & 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.  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), 1819-1833. 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, Vol. 1513, Berlin Heidelberg, Springer-Verlag, 1992.  Google Scholar

[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.  Google Scholar

[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.  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), 42-47.  Google Scholar

[9]

Y. N. Dowker and F. G. Friedlander, On limit sets in dynamical systems, Proc. London Math. Soc. (3), 4 (1954), 168-176.  Google Scholar

[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.  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), 821-828. 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, 216, Marcel Dekker, Inc., New York, 1999.  Google Scholar

[13]

M. Hurley, Chain recurrence and attraction in non-compact spaces, Ergod. Th. & Dynam. Sys., 11 (1991), 709-729. 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), 299-316. 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), 76-86. 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), 223-233. 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, 158, Marcel Dekker, New York, 1992.  Google Scholar

[19]

A. N. Sharkovsky, Continuous mapping on the limit points of an iteration sequence, (Russian) Ukrain. Mat. Zh., 18 (1966), 127-130.  Google Scholar

[20]

A. N. Sharkovsky, Partially ordered system of attracting sets, (Russian) Dokl. Akad. Nauk SSSR, 170 (1966), 1276-1278.  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), 6-9.  Google Scholar

show all references

References:
[1]

G. Acosta, A. Illanes and H. Méndez-Lango, The transitivity of induced maps, Topology Appl., 156 (2009), 1013-1033. 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), 1819-1833. 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, Vol. 1513, Berlin Heidelberg, Springer-Verlag, 1992.  Google Scholar

[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.  Google Scholar

[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.  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), 42-47.  Google Scholar

[9]

Y. N. Dowker and F. G. Friedlander, On limit sets in dynamical systems, Proc. London Math. Soc. (3), 4 (1954), 168-176.  Google Scholar

[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.  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), 821-828. 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, 216, Marcel Dekker, Inc., New York, 1999.  Google Scholar

[13]

M. Hurley, Chain recurrence and attraction in non-compact spaces, Ergod. Th. & Dynam. Sys., 11 (1991), 709-729. 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), 299-316. 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), 76-86. 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), 223-233. 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, 158, Marcel Dekker, New York, 1992.  Google Scholar

[19]

A. N. Sharkovsky, Continuous mapping on the limit points of an iteration sequence, (Russian) Ukrain. Mat. Zh., 18 (1966), 127-130.  Google Scholar

[20]

A. N. Sharkovsky, Partially ordered system of attracting sets, (Russian) Dokl. Akad. Nauk SSSR, 170 (1966), 1276-1278.  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), 6-9.  Google Scholar

[1]

Sergio Muñoz. Robust transitivity of maps of the real line. Discrete & Continuous Dynamical Systems, 2015, 35 (3) : 1163-1177. doi: 10.3934/dcds.2015.35.1163
[2]

Hiromichi Nakayama, Takeo Noda. Minimal sets and chain recurrent sets of projective flows induced from minimal flows on $3$-manifolds. Discrete & Continuous Dynamical Systems, 2005, 12 (4) : 629-638. doi: 10.3934/dcds.2005.12.629
[3]

Matthew Nicol. Induced maps of hyperbolic Bernoulli systems. Discrete & Continuous Dynamical Systems, 2001, 7 (1) : 147-154. doi: 10.3934/dcds.2001.7.147
[4]

Jun Hu, Oleg Muzician, Yingqing Xiao. Dynamics of regularly ramified rational maps: Ⅰ. Julia sets of maps in one-parameter families. Discrete & Continuous Dynamical Systems, 2018, 38 (7) : 3189-3221. doi: 10.3934/dcds.2018139
[5]

Suzanne Lynch Hruska. Rigorous numerical models for the dynamics of complex Hénon mappings on their chain recurrent sets. Discrete & Continuous Dynamical Systems, 2006, 15 (2) : 529-558. doi: 10.3934/dcds.2006.15.529
[6]

Arno Berger, Roland Zweimüller. Invariant measures for general induced maps and towers. Discrete & Continuous Dynamical Systems, 2013, 33 (9) : 3885-3901. doi: 10.3934/dcds.2013.33.3885
[7]

Dmitri Finkelshtein, Yuri Kondratiev, Yuri Kozitsky. Glauber dynamics in continuum: A constructive approach to evolution of states. Discrete & Continuous Dynamical Systems, 2013, 33 (4) : 1431-1450. doi: 10.3934/dcds.2013.33.1431
[8]

Gabriella Bretti, Ciro D’Apice, Rosanna Manzo, Benedetto Piccoli. A continuum-discrete model for supply chains dynamics. Networks & Heterogeneous Media, 2007, 2 (4) : 661-694. doi: 10.3934/nhm.2007.2.661
[9]

Christopher Cleveland. Rotation sets for unimodal maps of the interval. Discrete & Continuous Dynamical Systems, 2003, 9 (3) : 617-632. doi: 10.3934/dcds.2003.9.617
[10]

Evelyn Sander. Hyperbolic sets for noninvertible maps and relations. Discrete & Continuous Dynamical Systems, 1999, 5 (2) : 339-357. doi: 10.3934/dcds.1999.5.339
[11]

Boju Jiang, Jaume Llibre. Minimal sets of periods for torus maps. Discrete & Continuous Dynamical Systems, 1998, 4 (2) : 301-320. doi: 10.3934/dcds.1998.4.301
[12]

Francisco J. López-Hernández. Dynamics of induced homeomorphisms of one-dimensional solenoids. Discrete & Continuous Dynamical Systems, 2018, 38 (9) : 4243-4257. doi: 10.3934/dcds.2018185
[13]

Victor Ayala, Adriano Da Silva, Luiz A. B. San Martin. Control systems on flag manifolds and their chain control sets. Discrete & Continuous Dynamical Systems, 2017, 37 (5) : 2301-2313. doi: 10.3934/dcds.2017101
[14]

Lluís Alsedà, David Juher, Pere Mumbrú. Minimal dynamics for tree maps. Discrete & Continuous Dynamical Systems, 2008, 20 (3) : 511-541. doi: 10.3934/dcds.2008.20.511
[15]

Carlos Correia Ramos, Nuno Martins, Paulo R. Pinto. Escape dynamics for interval maps. Discrete & Continuous Dynamical Systems, 2019, 39 (11) : 6241-6260. doi: 10.3934/dcds.2019272
[16]

Tien-Cuong Dinh, Nessim Sibony. Rigidity of Julia sets for Hénon type maps. Journal of Modern Dynamics, 2014, 8 (3&4) : 499-548. doi: 10.3934/jmd.2014.8.499
[17]

Héctor E. Lomelí. Heteroclinic orbits and rotation sets for twist maps. Discrete & Continuous Dynamical Systems, 2006, 14 (2) : 343-354. doi: 10.3934/dcds.2006.14.343
[18]

Song Shao, Xiangdong Ye. Non-wandering sets of the powers of maps of a star. Discrete & Continuous Dynamical Systems, 2003, 9 (5) : 1175-1184. doi: 10.3934/dcds.2003.9.1175
[19]

Bruce Kitchens, Michał Misiurewicz. Omega-limit sets for spiral maps. Discrete & Continuous Dynamical Systems, 2010, 27 (2) : 787-798. doi: 10.3934/dcds.2010.27.787
[20]

Brenton LeMesurier. Continuum approximations for pulses generated by impulsive initial data in binary exciton chain systems. Discrete & Continuous Dynamical Systems - B, 2016, 21 (6) : 1813-1837. doi: 10.3934/dcdsb.2016024

2020 Impact Factor: 1.392

Tools

Metrics

  • PDF downloads (77)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences