On minimal sets of continuous maps on one-dimensional continua
https://doi.org/10.26907/2949-3919.2025.3.110-135
Abstract
Let X be a finite tree and let f : X → X be a continuous map with zero topological entropy and an infinite minimal set M. We show that the restriction of f|M of f to M is topologically conjugate to adding-machine τα, where α = (j1, . . . , jn, 2, 2, . . .) be the sequence for ji ≥ 2 if 1 ≤ i ≤ n. We describe the topological structure of finite trees on which there exist continuous maps with zero topological entropy and an infinite minimal set M on which the map f|M is topologically conjugate to adding machine τα, where α = (j1, . . . , jn, 2, 2, . . .). At the same time, for any sequence α = (j1, . . . , ji , . . .), where ji ≥ 2 for all i ≥ 1, there exist a dendrite X that is not a finite tree and a continuous map f with zero topological entropy and an infinite minimal set M on which the map f is topologically conjugate to adding machine τα.
We also show that for any sequence α = (j1, . . . , jn, . . .), where ji ≥ 2 for all i ≥ 1, there exist a dendrite X that is not a finite tree and a continuous map f with zero topological entropy and an infinite minimal set M such that f|M is topologically conjugate to adding-machine τα.
About the Author
E. N. Makhrova
Lobachevsky State University of Nizhny Novgorod
Russian Federation
Russian Federation
Elena Nikolaevna Makhrova
23 Gagarin ave., Nizhny Novgorod 603022
References
1. K. Kuratowski, Topology. Vol. 1, Academic Press, New York-London, 1966; Academic Press, New York-London, 1968. DOI: https://doi.org/10.1016/C2013-0-11022-7 https://doi.org/10.1016/C2013-0-11023-9
2. A.N. Sharkovsky, Yu.L. Maistrenko, E.Yu. Romanenko, Difference equations and their applications, Kluwer Academic Publishers Group, Dordrecht, 1993. DOI: https://doi.org/10.1007/978-94-011-1763-0
3. L. Alsed`a, X. Ye, No division and the set of periods for tree maps, Ergodic Theory Dynam. Systems 15 (2), 221–237 (1995). DOI: https://doi.org/10.1017/S0143385700008348
4. X. Ye, D-function of a minimal set and an extension of Sharkovskii’s theorem to minimal sets, Ergodic Theory Dynam. Systems 12 (2), 365–376 (1992). DOI: https://doi.org/10.1017/S0143385700006817
5. L.Alsed`a, X. Ye, Minimal sets of maps of Y , J. Math. Anal. Appl. 187 (1), 324–338 (1994). DOI: https://doi.org/10.1006/jmaa.1994.1359
6. B. Hasselblat, A. Katok, A first course in dynamics. With a panorama of recent developments, Cambridge University Press, New York, 2003.
7. L.S. Efremova, E.N. Makhrova, The dynamics of monotone maps of dendrites, Sb. Math. 192 (6), 807–821 (2001). DOI: https://doi.org/10.1070/SM2001v192n06ABEH000570
8. E.N. Makhrova, Remarks on minimal sets on dendrites and finite graphs, J. Difference Equ. Appl. 29 (9–12), 1313–1322 (2023). DOI: https://doi.org/10.1080/10236198.2023.2220417
9. L. Block, J. Keesling, A characterization of adding machine maps, Topology Appl. 140 (2–3), 151–161 (2004). DOI: https://doi.org/10.1016/j.topol.2003.07.006
10. J. Banks, Regular periodic decompositions for topologically transitive maps, Ergodic Theory Dynam. Systems 17 (3), 505–529 (1997). DOI: https://doi.org/10.1017/S0143385797069885
11. M.J. Feigenbaum, Universal behavior in nonlinear systems, Los Alamos Sci. 1 (1), 4–27 (1980). DOI: https://doi.org/10.1016/0167-2789(83)90112-4
12. Z. Nitecki, Topological dynamics on the interval, in: A. Katok, Ergodic theory and dynamical systems II, Progr. Math. 21, Birkh¨auser, Boston, MA, 1–73 (1982). https://doi.org/10.1007/978-1-4899-2689-0_1
13. A. Blokh, The “spectral” decomposition for one-dimensional maps, in: Dynam. Report. Expositions Dynam. Systems 4, 1–59 (1995). DOI: https://doi.org/10.1007/978-3-642-61215-2_1
14. F. Balibrea, R. Hric, L’. Snoha, Minimal sets on graphs and dendrites, Internat. J. Bifur. Chaos Appl. Sci. Engrg. 13 (7), 1721–1725 (2003). DOI: https://doi.org/10.1142/S0218127403007576
15. F. Balibrea, T. Downarowicz, R. Hric, L’. Snoha, V. Spitalsk´y, ˘ Almost totally disconnected minimal systems, Ergodic Theory Dynam. Systems 29 (3), 737–766 (2009). DOI: https://doi.org/10.1017/S0143385708000540
16. H.-O. Peitgen, P.H. Richter, The beauty of fractals. Images of complex dynamical systems, Springer-Verlag, Berlin, 1986. DOI: https://doi.org/10.1007/978-3-642-61717-1
17. S.J. Agronsky, J.G. Ceder, What sets can be ω-limit sets in E n?, Real Anal. Exchange 17 (1), 97–109 (1991/1992). DOI: https://doi.org/10.2307/44152199
18. F. Balibrea, J.L. Garc´ia-Guirao, Continua with empty interior as ω-limit sets, Appl. Gen. Topol. 6 (2), 195–205 (2005).
19. L.S. Efremova, Example of the smooth skew product in the plane with the one-dimensional ramified continuum as the global attractor, in: European Conference on Iteration Theory 2010, ESAIM Proc. 36, EDP Sci., Les Ulis, 15–25 (2012). DOI: https://doi.org/10.1051/proc/201236002
20. L.S. Efremova, Ramified continua as global attractors of C 1 -smooth self-maps of a cylinder close to skew products, J. Difference Equ. Appl. 29 (9–12), 1244–1274 (2023). DOI: https://doi.org/10.1080/10236198.2023.2204144
21. D. Drozdov, A. Tetenov, On the classification of fractal square dendrites, Adv. Theory Nonlinear Anal. Appl. 7 (3), 19–96 (2023). DOI: https://doi.org/10.17762/atnaa.v7.i3.276
22. V.Zh. Sakbaev, O.G. Smolyanov, Diffusion and quantum dynamics on graphs, Dokl. Math. 88 (1), 404–408 (2013). DOI: https://doi.org/10.1134/S1064562413040108
23. L.S. Efremova, E.N. Makhrova, One-dimensional dynamical systems, Russian Math. Surveys 76 (5), 821–881 (2021). DOI: https://doi.org/10.1070/RM9998
24. G.D. Birkhoff, Dynamical systems, AMS, New York, 1927.
25. R. Munasinghe, Composants, unstable sets, and minimal sets of inverse limit spaces, Thesis (Ph.D.), University of Wyoming. ProQuest LLC, Ann Arbor, MI, 1992.
26. S. Kolyada, L’. Snoha, Minimal dynamical systems, Scholarpedia 4 (11), art. 5803. URL: https://www.scholarpedia.org/article/Minimal$_$dynamical$_$systems
27. R.L. Adler, A.G. Konheim, M.H. McAndrew, Topological entropy, Trans. Amer. Math. Soc. 114, 309–319 (1965). DOI: https://doi.org/10.1090/S0002-9947-1965-0175106-9
28. E.N. Makhrova, The structure of dendrites with the periodic point property, Russian Math. (Iz. VUZ) 55 (11) 33–37 (2011). DOI: https://doi.org/10.3103/S1066369X11110053
29. E.N. Makhrova, Remarks on the existence of periodic points for continuous maps on dendrites, Lobachevskii J. Math. 43 (7), 1711–1719 (2022). DOI: https://doi.org/10.1134/S1995080222100274
30. G.R. Gordon Jr., L. Lum, Monotone retracts and some characterizations of dendrites, Proc. Amer. Math. Soc. 59 (1), 156–158 (1976). DOI: https://doi.org/10.1090/S0002-9939-1976-0423317-X
31. Yu.S. Barkovskii, G.M. Levin, A Cantor limit set, Russ. Math. Surveys 35(2), 235–236 (1980). DOI: https://doi.org/10.1070/RM1980v035n02ABEH001644
32. H.M. Gehman, Concerning the subsets of a plane continuous curve, Ann. Math. 27 (1), 29–46 (1925). DOI: https://doi.org/10.2307/1967832
Review
For citations:
Makhrova E.N. On minimal sets of continuous maps on one-dimensional continua. Mathematics and Theoretical Computer Science. 2025;3(3):110-135. (In Russ.) https://doi.org/10.26907/2949-3919.2025.3.110-135
Views:
17
Email this article 