matatecsМатематика и теоретические компьютерные наукиMathematics and Theoretical Computer Science2949-3919Казанский (Приволжский) федеральный университет10.26907/2949-3919.2025.3.110-135matatecs-89Research ArticleСТАТЬИО минимальных множествах непрерывных отображений на одномерных континуумахOn minimal sets of continuous maps on one-dimensional continuaМахроваЕ. Н.MakhrovaE. N.

Елена Николаевна Махровапр-т Гагарина, д. 23, г. Нижний Новгород, 603022

Elena Nikolaevna Makhrova23 Gagarin ave., Nizhny Novgorod 603022

elena_makhrova@inbox.ruНижегородский государственный университет им. Н.И. ЛобачевскогоРоссияLobachevsky State University of Nizhny NovgorodRussian Federation20252910202533110135Copyright © Махрова Е.Н., 20252025Махрова Е.Н.Makhrova E.N.Данная работа распространяется под лицензией Creative Commons Attribution 4.0.This work is licensed under a Creative Commons Attribution 4.0 License.https://matatecs.elpub.ru/jour/article/view/89

Пусть X – конечное дерево, а f : X → X – непрерывное отображение, имеющее нулевую топологическую энтропию и бесконечное минимальное множество M. Нами доказано, что сужение f|M отображения f на M топологически сопряжено отображению счетчика τα, где α = (j1, . . . , jn, 2, 2, . . .) есть последовательность при ji ≥ 2 для 1 ≤ i ≤ n. Дано описание топологической структуры конечных деревьев, на которых существуют непрерывные отображения с нулевой топологической энтропией и бесконечным минимальным множеством M, на котором отображение f|M топологически сопряжено счетчику τα, где α = (j1, . . . , jn, 2, 2, . . .). В то же время для любой последовательности α = (j1, . . . , ji , . . .), где ji ≥ 2 для всех i ≥ 1, существуют дендрит X, не являющийся конечным деревом, и непрерывное отображение f с нулевой топологической энтропией и бесконечным минимальным множеством M, на котором отображение f топологически сопряжено счетчику τα.

Нами также показано: если X – дендрит, а f : X → X – непрерывное отображение, имеющее нулевую топологическую энтропию и бесконечное минимальное множество M, то существует такая последовательность α = (j1, . . . , ji , . . .) (ji ≥ 2), что отображение f|M полусопряжено отображению счетчика τα. 

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. 

дендритконечное деревоминимальное множествоотображение счетчика (adding-machine)топологическая энтропияпочти периодическая точкарекуррентная точкаdendritefinite treeminimal setadding-machinetopological entropyalmost periodic pointrecurrent pointИсследование выполнено за счет гранта Российского научного фонда № 24-21-00242, https://rscf.ru/project/24-21-00242/.ReferencesК. Куратовский, Топология, Т. 1, Мир, М., 1966; Т. 2, Мир, М., 1969.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/C2013-0-11023-9А.Н.Шарковский, Ю.Л. Майстренко, Е.Ю. Романенко, Разностные уравнения и их приложения, Наук. думка, Киев, 1986.A.N. Sharkovsky, Yu.L. Maistrenko, E.Yu. Romanenko, Difference equations and their applications, Kluwer Academic Publishers Group, Dordrecht, 1993. DOI: https://doi.org/10.1007/978-94-011-1763-0L. 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/S0143385700008348L. 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/S0143385700008348X. 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/S0143385700006817X. 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/S0143385700006817L.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.1994.1359L.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.1994.1359B. Hasselblat, A. Katok, A first course in dynamics. With a panorama of recent developments, Cambridge University Press, New York, 2003.B. Hasselblat, A. Katok, A first course in dynamics. With a panorama of recent developments, Cambridge University Press, New York, 2003.Л.С. Ефремова, Е.Н. Махрова, Динамика монотонных отображений дендритов, Матем. сб. 192 (6), 15–30 (2001). DOI: https://doi.org/10.4213/sm570L.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/SM2001v192n06ABEH000570E.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/10.1080/10236198.2023.2220417E.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/10.1080/10236198.2023.2220417L. 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.2003.07.006L. 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.2003.07.006J. Banks, Regular periodic decompositions for topologically transitive maps, Ergodic Theory Dynam. Systems 17 (3), 505–529 (1997). DOI: https://doi.org/10.1017/S0143385797069885J. Banks, Regular periodic decompositions for topologically transitive maps, Ergodic Theory Dynam. Systems 17 (3), 505–529 (1997). DOI: https://doi.org/10.1017/S0143385797069885М. Фейгенбаум, Универсальность в поведении нелинейных систем, УФН 141 (2), 343–374 (1983). DOI: https://doi.org/10.3367/UFNr.0141.198310e.0343M.J. Feigenbaum, Universal behavior in nonlinear systems, Los Alamos Sci. 1 (1), 4–27 (1980). DOI: https://doi.org/10.1016/0167-2789(83)90112-4Z. 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_1Z. 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_1A. 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_1A. 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_1F. 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/S0218127403007576F. 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/S0218127403007576F. 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/S0143385708000540F. 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/S0143385708000540H.-O. Peitgen, P.H. Richter, The beauty of fractals. Images of complex dynamical systems, Springer-Verlag, Berlin, 1986. DOI: https://doi.org/10.1007/978-3-642-61717-1H.-O. Peitgen, P.H. Richter, The beauty of fractals. Images of complex dynamical systems, Springer-Verlag, Berlin, 1986. DOI: https://doi.org/10.1007/978-3-642-61717-1S.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/10.2307/44152199S.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/10.2307/44152199F. Balibrea, J.L. Garc´ia-Guirao, Continua with empty interior as ω-limit sets, Appl. Gen. Topol. 6 (2), 195–205 (2005).F. Balibrea, J.L. Garc´ia-Guirao, Continua with empty interior as ω-limit sets, Appl. Gen. Topol. 6 (2), 195–205 (2005).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/201236002L.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/201236002L.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/10.1080/10236198.2023.2204144L.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/10.1080/10236198.2023.2204144D. 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.i3.276D. 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.i3.276В.Ж. Сакбаев, О.Г. Смолянов, Диффузия и квантовая динамика на графах, Докл. РАН, 451 (2), 141–145 (2013). DOI: https://doi.org/10.7868/s0869565213200061V.Zh. Sakbaev, O.G. Smolyanov, Diffusion and quantum dynamics on graphs, Dokl. Math. 88 (1), 404–408 (2013). DOI: https://doi.org/10.1134/S1064562413040108Л.С. Ефремова, Е.Н. Махрова, Одномерные динамические системы, УМН 76 (5), 81–146 (2021). DOI: https://doi.org/10.4213/rm9998L.S. Efremova, E.N. Makhrova, One-dimensional dynamical systems, Russian Math. Surveys 76 (5), 821–881 (2021). DOI: https://doi.org/10.1070/RM9998Дж.Д. Биркгоф, Динамические системы, Регулярная и хаотическая динамика, Москва–Ижевск, 2002.G.D. Birkhoff, Dynamical systems, AMS, New York, 1927.R. Munasinghe, Composants, unstable sets, and minimal sets of inverse limit spaces, Thesis (Ph.D.), University of Wyoming. ProQuest LLC, Ann Arbor, MI, 1992.R. Munasinghe, Composants, unstable sets, and minimal sets of inverse limit spaces, Thesis (Ph.D.), University of Wyoming. ProQuest LLC, Ann Arbor, MI, 1992.S. Kolyada, L’. Snoha, Minimal dynamical systems, Scholarpedia 4 (11), art. 5803. URL: https://www.scholarpedia.org/article/Minimal$_$dynamical$_$systemsS. Kolyada, L’. Snoha, Minimal dynamical systems, Scholarpedia 4 (11), art. 5803. URL: https://www.scholarpedia.org/article/Minimal$_$dynamical$_$systemsR.L. Adler, A.G. Konheim, M.H. McAndrew, Topological entropy, Trans. Amer. Math. Soc. 114, 309–319 (1965). DOI: https://doi.org/10.1090/S0002-9947-1965-0175106-9R.L. Adler, A.G. Konheim, M.H. McAndrew, Topological entropy, Trans. Amer. Math. Soc. 114, 309–319 (1965). DOI: https://doi.org/10.1090/S0002-9947-1965-0175106-9Е.Н. Махрова, Структура дендритов со свойством существования периодических точек, Изв. вузов. Матем. (11), 41–45 (2011). URL: https://www.mathnet.ru/ivm8393E.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/S1066369X11110053E.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/S1995080222100274E.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/S1995080222100274G.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-XG.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-XЮ.С. Барковский, Г.М. Левин, О предельном канторовом множестве, УМН 35 (2), 201–202 (1980). URL: https://www.mathnet.ru/rus/rm3269Yu.S. Barkovskii, G.M. Levin, A Cantor limit set, Russ. Math. Surveys 35(2), 235–236 (1980). DOI: https://doi.org/10.1070/RM1980v035n02ABEH001644H.M. Gehman, Concerning the subsets of a plane continuous curve, Ann. Math. 27 (1), 29–46 (1925). DOI: https://doi.org/10.2307/1967832H.M. Gehman, Concerning the subsets of a plane continuous curve, Ann. Math. 27 (1), 29–46 (1925). DOI: https://doi.org/10.2307/1967832

The authors declare that there are no conflicts of interest present.