Abundance of Fast Growth of the Number of Periodic Points in 2-Dimensional Area-Preserving Dynamics

1. Arnold, V.I.: Arnolds problems. Translated and revised edition of the 2000 Russian original. With a preface by Philippov, V., Yakivchik, A., Peters, M. Springer, Berlin; PHASIS, Moscow (2004)

2. Artin M., Mazur B.: On periodic points. Ann. Math. 81(1), 82–99 (1965)

   Article  MathSciNet  MATH  Google Scholar 

3. Asaoka, M., Turaev, D., Shinohara, K.: Degenerate behavior in non-hyperbolic semigroup actions on the interval: fast growth of periodic points and universal dynamics. Math. Ann. (2016). doi:10.1007/s00208-016-1468-0

4. Berger P.: Generic family with robustly infinitely many sinks. Invent. Math. 205, 121–172 (2016)

   Article  ADS  MathSciNet  MATH  Google Scholar 

5. Berger, P.: Generic family displaying robustly a fast growth of the number of periodic points. Preprint. arXiv:1701.0239

6. Broer H.W., Tangerman F.M.: From a differentiable to a real analytic perturbation theory, applications to the Kupka Smale theorems. Ergod. Theory Dyn. Syst. 6, 345–362 (1986)

   Article  MathSciNet  MATH  Google Scholar 

7. Bonatti C., Díaz L., Fisher T.: Super-exponential growth of the number of periodic points inside homoclinic classes. Discrete Contin. Dyn. Syst. 20(3), 589–604 (2008)

   MathSciNet  MATH  Google Scholar 

8. de Melo W.: A finiteness problem for one dimensional maps. Proc. AMS 101, 721–727 (1987)

   Article  MathSciNet  MATH  Google Scholar 

9. Hunt, B., Kaloshin, V.Yu.: Prevalence. In: Handbook of Dynamical Systems, vol. 3, pp. 43–87. Elsevier/North-Holland, Amsterdam (2010)

10. Hunt B.R., Sauer T., Yorke J.A.: Prevalence: a translation-invariant “almost every” on infinite-dimensional spaces. Bull. Amer. Math. Soc. (N.S.) 27(2), 217–238 (1992)

    Article  MathSciNet  MATH  Google Scholar 

11. Ilyashenko, Y., Li, W.: Nonlocal bifurcations. Mathematical Surveys and Monographs, 66. American Mathematical Society, Providence, RI, xiv+286 pp (1999)

12. Kaloshin V.Y.: Some prevalent properties of smooth dynamical systems. Proc. Steklov Inst. Math. 213, 123–151 (1997)

    MathSciNet  Google Scholar 

13. Kaloshin V.Y.: An extension of the Artin–Mazur theorem. Ann. Math. (2) 150(2), 729–741 (1999)

    Article  MathSciNet  MATH  Google Scholar 

14. Kaloshin V.: Generic diffeomorphisms with superexponential growth of number of periodic orbits. Commun. Math. Phys. 211(1), 253–271 (2000)

    Article  ADS  MathSciNet  MATH  Google Scholar 

15. Kaloshin V.Yu., Hunt B.: A stretched exponential bound on the rate of growth of the number of periodic points for prevalent diffeomorphisms I. Electron. Res. Announc. Am. Math. Soc. 7, 17–27 (2001)

    Article  MathSciNet  MATH  Google Scholar 

16. Kaloshin V.Yu., Hunt B.: A stretched exponential bound on the rate of growth of the number of periodic points for prevalent diffeomorphisms II. Electron. Res. Announc. Am. Math. Soc. 7, 28–36 (2001) MR1826993

    Article  MathSciNet  MATH  Google Scholar 

17. Kaloshin V.Yu., Hunt B.: Stretched exponential estimates on grow of the number of periodic points for prevalent diffeomorphisms. Ann. Math. 165, 89–170 (2007)

    Article  MathSciNet  MATH  Google Scholar 

18. Kaloshin V., Kozlovski O.S.: A C ^r unimodal map with an arbitrary fast growth of the number of periodic points. Ergod. Theory Dyn. Syst. 32(1), 159–165 (2012)

    Article  MathSciNet  MATH  Google Scholar 

19. Kaloshin V., Saprykina M.: Generic 3-dimensional volume-preserving diffeomorphisms with superexponential growth of number of periodic orbits. Discrete Contin. Dyn. Syst. 15(2), 611–640 (2006)

    Article  MathSciNet  MATH  Google Scholar 

20. Kozlovski, O.S.: The dynamics of intersections of analytic manifolds. (Russian) Dokl. Akad. Nauk 323 (1992), no. 5, 823–825; translation in Russian Acad. Sci. Dokl. Math. 45 (1992), no. 2, 425–427 (1993)

21. Krantz, S.G., Park, H.R.: A primer of real analytic functions, 2nd edn. Birkhäuser Advanced Texts: Basler Lehrbücher, Birkhäuser Boston, Inc., Boston, MA (2002)

22. Martens M., de Melo W., van Strien S.: Julia–Fatou–Sullivan theory for real one-dimensional dynamics. Acta Math. 168(3–4), 273–318 (1992)

    Article  MathSciNet  MATH  Google Scholar 

23. Moser, J.K.: Stable and random motions in dynamical systems. Reprint of the 1973 original. Princeton University Press, Princeton, NJ, xii+198 pp (2001)

24. Newhouse S.E.: Diffeomorphisms with infinitely many sinks. Topology 12, 9–18 (1974)

    Article  MathSciNet  MATH  Google Scholar 

25. Palis J.: A global perspective for non-conservative dynamics. Ann. Inst. H. Poincaré Anal. Non Linéaire 22, 485–507 (2005)

    Article  ADS  MathSciNet  MATH  Google Scholar 

26. Robinson, C.: Dynamical Systems. Stability, Symbolic Dynamics, and Chaos, 2nd edn. Studies in Advanced Mathematics. CRC Press, Boca Raton, FL (1999)

27. Siegel, C.L., Moser J.K.: Lectures on celestial mechanics. Translation by Charles I. Kalme. Die Grundlehren der mathematischen Wissenschaften, Band 187. Springer–Verlag, New York–Heidelberg (1971)

28. Takens, F.: A non-stabilizable jet of a singularity of a vector field: the analytic case. Algebraic and differential topology–global differential geometry, 288–305, Teubner-Texte Math., 70, Teubner, Leipzig (1984)

29. Ott W., Yorke J.Y.: Prevalence. Bull. Am. Math. Soc. (N.S.) 42(3), 263–290 (2005)

    Article  MathSciNet  MATH  Google Scholar 

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