A. S. Gorodetski, Yu. S. Ilyashenko, “Certain Properties of Skew Products over a Horseshoe and a Solenoid”, Dynamical systems, automata, and infinite groups, Collected papers, Trudy Mat. Inst. Steklova, 231, Nauka, MAIK «Nauka/Inteperiodika», M., 2000, 96–118; Proc. Steklov Inst. Math., 231 (2000), 90

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Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2000, Volume 231, Pages 96–118 (Mi tm513)

This article is cited in 34 scientific papers (total in 35 papers)

Certain Properties of Skew Products over a Horseshoe and a Solenoid

A. S. Gorodetski, Yu. S. Ilyashenko

Full-text PDF (322 kB) Citations (35)

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Abstract: The skew products are investigated over the Bernoulli shift and the Smale–Williams solenoid with a fiber

$S^1$ . It is assumed that the mapping in the fiber Hölder continuously depends on a point in the base (it is these skew products that arise in the study of partially hyperbolic sets). It is proved that, in the space of skew products with this property, there exists an open domain such that the mappings from this domain have dense sets of periodic orbits that are attracting and repelling along the fiber, as well as the dense orbits with the zero (along the fiber) Lyapunov exponent.

Received in February 2000

Bibliographic databases:

Document Type: Article

UDC: 517.938

Language: Russian

Citation: A. S. Gorodetski, Yu. S. Ilyashenko, “Certain Properties of Skew Products over a Horseshoe and a Solenoid”, Dynamical systems, automata, and infinite groups, Collected papers, Trudy Mat. Inst. Steklova, 231, Nauka, MAIK «Nauka/Inteperiodika», M., 2000, 96–118; Proc. Steklov Inst. Math., 231 (2000), 90–112

Citation in format AMSBIB

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\by A.~S.~Gorodetski, Yu.~S.~Ilyashenko

\paper Certain Properties of Skew Products over a~Horseshoe and a~Solenoid

\inbook Dynamical systems, automata, and infinite groups

\bookinfo Collected papers

\serial Trudy Mat. Inst. Steklova

\yr 2000

\vol 231

\pages 96--118

\publ Nauka, MAIK «Nauka/Inteperiodika»

\publaddr M.

\mathnet{http://mi.mathnet.ru/tm513}

\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=1841753}

\zmath{https://zbmath.org/?q=an:1002.37017}

\transl

\jour Proc. Steklov Inst. Math.

\yr 2000

\vol 231

\pages 90--112

Linking options:

https://www.mathnet.ru/eng/tm513

https://www.mathnet.ru/eng/tm/v231/p96

This publication is cited in the following 35 articles:

T. N. Bakiev, A. I. Bufetov, V. A. Vasilev, S. M. Voronin, A. A. Glutsyuk, A. S. Gorodetskii, A. V. Dukov, V. Yu. Kaloshin, A. V. Klimenko, V. V. Kozlov, S. B. Kuksin, S. K. Lando, V. S. Oganesyan, G. I. Olshanskii, S. Yu. Pilyugin, O. V. Pochinka, Ya. G. Sinai, A. S. Skripchenko, A. L. Skubachevskii, I. A. Taimanov, V. A. Timorin, V. M. Tikhomirov, D. V. Treschev, D. A. Filimonov, K. M. Khanin, Kh. Khedenmalm, A. G. Khovanskii, M. A. Tsfasman, A. I. Shafarevich, I. S. Shilin, S. Yu. Yakovenko, “K 80-letiyu Yuliya Sergeevicha Ilyashenko”, UMN, 80:2(482) (2025), 171–183
Diaz L.J. Gelfert K. Rams M., “Entropy Spectrum of Lyapunov Exponents For Nonhyperbolic Step Skew-Products and Elliptic Cocycles”, Commun. Math. Phys., 367:2 (2019), 351–416
Stavros Anastassiou, Anastasios Bountis, Arnd Bäcker, “Recent Results on the Dynamics of Higher-dimensional Hénon Maps”, Regul. Chaotic Dyn., 23:2 (2018), 161–177
L. S. Efremova, “Dynamics of skew products of interval maps”, Russian Math. Surveys, 72:1 (2017), 101–178
Das S., Yorke J.A., “Multichaos From Quasiperiodicity”, SIAM J. Appl. Dyn. Syst., 16:4 (2017), 2196–2212
Barrientos P.G., Fakhari A., Malicet D., Sarizadeh A., “Expanding Actions: Minimality and Ergodicity”, Stoch. Dyn., 17:4 (2017), 1750031
Asaoka M., Shinohara K., Turaev D., “Degenerate Behavior in Non-Hyperbolic Semigroup Actions on the Interval: Fast Growth of Periodic Points and Universal Dynamics”, Math. Ann., 368:3-4 (2017), 1277–1309
Ilyashenko Yu., Shilin I., “Attractors and Skew Products”, Modern Theory of Dynamical Systems: a Tribute to Dmitry Victorovich Anosov, Contemporary Mathematics, 692, eds. Katok A., Pesin Y., Hertz F., Amer Mathematical Soc, 2017, 155–175
Ilyashenko Yu., Romaskevich O., “Sternberg Linearization Theorem for Skew Products”, J. Dyn. Control Syst., 22:3 (2016), 595–614
J. Math. Sci. (N. Y.), 209:6 (2015), 979–987
Tatiana Golenishcheva-Kutuzova, Anton Gorodetski, Victor Kleptsyn, Denis Volk, “Translation numbers define generators of $F_k^+\to\mathrm{Homeo}_+(\mathbb S^1)$ ”, Mosc. Math. J., 14:2 (2014), 291–308
Homburg A.J., Nassiri M., “Robust Minimality of Iterated Function Systems With Two Generators”, Ergod. Theory Dyn. Syst., 34:6 (2014), 1914–1929
Gonchenko S.V. Gonchenko A.S. Ovsyannikov I.I. Turaev D.V., “Examples of Lorenz-Like Attractors in Henon-Like Maps”, Math. Model. Nat. Phenom., 8:5 (2013), 48–70
Kryzhevich S., Wiercigroch M., “Topology of Vibro-Impact Systems in the Neighborhood of Grazing”, Physica D, 241:22 (2012), 1919–1931
Ghane F.H., Nazari M., Saleh M., Shabani Z., “Attractors and their Invisible Parts for Skew Products with High Dimensional Fiber”, Int. J. Bifurcation Chaos, 22:8 (2012), 1250182
Ilyashenko Yu., Negut A., “Holder Properties of Perturbed Skew Products and Fubini Regained”, Nonlinearity, 25:8 (2012), 2377–2399
Diaz L.J., Gelfert K., “Porcupine-Like Horseshoes: Transitivity, Lyapunov Spectrum, and Phase Transitions”, Fundam. Math., 216:1 (2012), 55–100
Homburg A.J., “Circle Diffeomorphisms Forced by Expanding Circle Maps”, Ergod. Theory Dyn. Syst., 32:Part 6 (2012), 2011–2024
Pilyugin S.Yu., “Theory of pseudo-orbit shadowing in dynamical systems”, Differ Equ, 47:13 (2011), 1929–1938
Ilyashenko Yu., “Thick attractors of boundary preserving diffeomorphisms”, Indag Math (N S ), 22:3–4 (2011), 257–314

Citing articles in Google Scholar: Russian citations, English citations
Related articles in Google Scholar: Russian articles, English articles

Труды Математического института имени В. А. Стеклова

Proceedings of the Steklov Institute of Mathematics

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