General information
Latest issue
Forthcoming papers
Impact factor
Guidelines for authors
License agreement
Submit a manuscript

Search papers
Search references

Latest issue
Current issues
Archive issues
What is RSS

Mat. Sb.:

Personal entry:
Save password
Forgotten password?

Mat. Sb. (N.S.), 1971, Volume 85(127), Number 2(6), Pages 163–188 (Mi msb3190)  

This article is cited in 30 scientific papers (total in 30 papers)

On smooth mappings of the circle into itself

M. V. Jakobson

Abstract: In this article is constructed the set $\mathfrak M=\mathfrak M_1\cup\mathfrak M_2$, open and everywhere dense in $C^1(S^1,S^1)$, of $\Omega$-stable mappings. $\Omega(f)$ is totally disconnected and $f/\Omega(f)$ is topologically conjugate to the topological Markov chain with a finite number of states; for $f\in\mathfrak M_2$ we have $\Omega(f)=S^1$ and $f/S^1$ topologically conjugate to $z^n/S^1$. For $f\in\mathfrak M$ there exists a hyperbolic structure onЁ$\Omega(f)$.
Figures: 1
Bibliography: 9 titles.

Full text: PDF file (3188 kB)
References: PDF file   HTML file

English version:
Mathematics of the USSR-Sbornik, 1971, 14:2, 161–185

Bibliographic databases:

UDC: 513.838
MSC: 58C25
Received: 15.04.1970

Citation: M. V. Jakobson, “On smooth mappings of the circle into itself”, Mat. Sb. (N.S.), 85(127):2(6) (1971), 163–188; Math. USSR-Sb., 14:2 (1971), 161–185

Citation in format AMSBIB
\by M.~V.~Jakobson
\paper On smooth mappings of the circle into itself
\jour Mat. Sb. (N.S.)
\yr 1971
\vol 85(127)
\issue 2(6)
\pages 163--188
\jour Math. USSR-Sb.
\yr 1971
\vol 14
\issue 2
\pages 161--185

Linking options:

    SHARE: FaceBook Twitter Livejournal

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

    This publication is cited in the following articles:
    1. M. V. Yakobson, “O svoistvakh odnoarametricheskogo semeistva dinamicheskikh sistem $x\mapsto A\cdot x\cdot e^{-x}$”, UMN, 31:2(188) (1976), 239–240  mathnet  mathscinet  zmath
    2. Lai-Sang Young, “A closing lemma on the interval”, Invent math, 54:2 (1979), 179  crossref  mathscinet  zmath  adsnasa
    3. John Guckenheimer, “Sensitive dependence to initial conditions for one dimensional maps”, Comm Math Phys, 70:2 (1979), 133  crossref  mathscinet  zmath
    4. M. V. Jakobson, “Invariant measures that are absolutely continuous in $ dx$ for one-parameter families of one-dimensional maps”, Russian Math. Surveys, 35:4 (1980), 198–199  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    5. P. Coullet, C. Tresser, A. Arneodo, “Transition to turbulence for doubly periodic flows”, Physics Letters A, 77:5 (1980), 327  crossref
    6. Carlos Arteaga, “Endomorphisms of Branche one-dimensional Manifolds”, Bol. Soc. Bras. Mat, 13:1 (1982), 93  crossref
    7. Stellan Ostlund, David Rand, James Sethna, Eric Siggia, “Universal properties of the transition from quasi-periodicity to chaos in dissipative systems”, Physica D: Nonlinear Phenomena, 8:3 (1983), 303  crossref
    8. Ricardo Mañé, “Hyperbolicity, sinks and measure in one dimensional dynamics”, Commun.Math. Phys, 100:4 (1985), 495  crossref
    9. H Nusse, “Persistence of order and structure in chaos”, Physica D: Nonlinear Phenomena, 20:2-3 (1986), 374  crossref
    10. H.E Nusse, “Qualitative analysis of the dynamics and stability properties for Axiom A maps”, Journal of Mathematical Analysis and Applications, 136:1 (1988), 74  crossref
    11. Steve Smale, “Dynamics retrospective: great problems, attempts that failed”, Physica D: Nonlinear Phenomena, 51:1-3 (1991), 267  crossref
    12. Gonzalo Contreras, “On the C2-creation of links of critical points”, Ergod Th Dynam Sys, 13:2 (1993)  crossref  mathscinet  zmath
    13. L. S. Efremova, “A class of twisted products of maps of an interval”, Math. Notes, 54:3 (1993), 890–898  mathnet  crossref  mathscinet  zmath  isi
    14. V. N. Belykh, “Chaotic and strange attractors of a two-dimensional map”, Sb. Math., 186:3 (1995), 311–326  mathnet  crossref  mathscinet  zmath  isi
    15. Steve Smale, “Mathematical problems for the next century”, Math Intelligencer, 20:2 (1998), 7  crossref  mathscinet  zmath
    16. E. V. Zhuzhoma, V. S. Medvedev, “The Closure Lemma for Piecewise Diffeomorphic Maps of the Circle”, Math. Notes, 69:2 (2001), 277–280  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    17. L. S. Efremova, “$\Omega$-Stable Skew Products of Interval Maps Are Not Dense in $T^1(I)$”, Proc. Steklov Inst. Math., 236 (2002), 157–163  mathnet  mathscinet  zmath
    18. Samuil Aranson, Mikhail Malkin, Vladislav Medvedev, Evgeny Zhuzhoma, “Versions of the closing lemma for certain dynamical systems on tori”, Qual Th Dyn Syst, 4:1 (2003), 1  crossref  mathscinet  zmath
    19. J. Palis, “A global perspective for non-conservative dynamics”, Annales de l'Institut Henri Poincare (C) Non Linear Analysis, 22:4 (2005), 485  crossref
    20. M. V. Shamolin, “Dynamical systems with variable dissipation: Approaches, methods, and applications”, J. Math. Sci., 162:6 (2009), 741–908  mathnet  crossref  mathscinet  zmath  elib  elib
    21. L. S. Efremova, “Space of $C^1$-smooth skew products of maps of an interval”, Theoret. and Math. Phys., 164:3 (2010), 1208–1214  mathnet  crossref  crossref  adsnasa  isi
    22. E. V. Zhuzhoma, N. V. Isaenkova, “Zero-dimensional solenoidal base sets”, Sb. Math., 202:3 (2011), 351–372  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    23. E. V. Zhuzhoma, N. V. Isaenkova, “Classification of coverings of the circle”, Proc. Steklov Inst. Math., 278 (2012), 88–93  mathnet  crossref  mathscinet  isi  elib  elib
    24. L. S. Efremova, “A decomposition theorem for the space of $C^1$-smooth skew products with complicated dynamics of the quotient map”, Sb. Math., 204:11 (2013), 1598–1623  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    25. Matheus C. Moreira C.G. Pujals E.R., “Axiom a Versus Newhouse Phenomena for Benedicks-Carleson Toy Models”, Ann. Sci. Ec. Norm. Super., 46:6 (2013), 857–878  isi
    26. L. S. Efremova, “Multivalued functions and nonwandering set of skew products of maps of an interval with complicated dynamics of quotient map”, Russian Math. (Iz. VUZ), 60:2 (2016), 77–81  mathnet  crossref  isi
    27. Efremova L.S., “Stability as a Whole of a Family of Fibers Maps and -Stability of C ^{1} -Smooth Skew Products of Maps of an Interval”, Noma15 International Workshop on Nonlinear Maps and Applications, Journal of Physics Conference Series, 692, ed. Gelfreich V. FournierPrunaret D. LopezRuiz R. Callegari S. Nishio Y. Blokhina E., IOP Publishing Ltd, 2016, 012010  crossref  isi  scopus
    28. V. Z. Grines, E. D. Kurenkov, “O strukture odnomernykh bazisnykh mnozhestv endomorfizmov poverkhnostei”, Zhurnal SVMO, 18:2 (2016), 16–24  mathnet  elib
    29. L. S. Efremova, “Dynamics of skew products of interval maps”, Russian Math. Surveys, 72:1 (2017), 101–178  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    30. V. Z. Grines, E. D. Kurenkov, “O giperbolicheskikh attraktorakh i repellerakh endomorfizmov”, Nelineinaya dinam., 13:4 (2017), 557–571  mathnet  crossref  elib
  • Математический сборник (новая серия) - 1964–1988 Sbornik: Mathematics (from 1967)
    Number of views:
    This page:406
    Full text:117

    Contact us:
     Terms of Use  Registration  Logotypes © Steklov Mathematical Institute RAS, 2020