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Discontinuity, Nonlinearity, and Complexity 9(4) (2020) 519--523 | DOI:10.5890/DNC.2020.12.004

L.S. Efremova

Institute of Information Technologies, Mathematics and Mechanics, Nizhni Novgorod State University, Nizhni Novgorod, 603950, Russia Department of General Mathematics, Moscow Institute of Physics and Technology, Moscow Region, Dolgoprudny, 141701, Russia

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Abstract

We study $C^1$-smooth maps obtained by small perturbations of $C^1$-smooth skew products of maps of an interval with $\Omega$-stable quotients and present results on the coexistence of periods of periodic orbits for maps under consideration. In particular, $C^1$-smooth $\Omega$-stable maps of an interval do not contain maps of type $2^{\infty}$, i.e. maps that have the unbounded set of (the least) periods of periodic orbits $\tau$ for $\tau=\{2^i\}_{i\geq 0}$. We prove here that analogously to $C^1$-smooth skew products of maps of an interval with $\Omega$-stable quotients there exist the maps under consideration with $\tau=\{2^i\}_{i\geq 0}.$

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