Evolution of the Sharkovsky theorem

Alexander Blokh; Michał Misiurewicz

doi:10.3842/umzh.v76i1.7641

Alexander Blokh Department of Mathematics, University of Alabama at Birmingham, USA
Michał Misiurewicz Department of Mathematical Sciences, Indiana University – Purdue University Indianapolis, USA

DOI: https://doi.org/10.3842/umzh.v76i1.7641

Keywords: the Sharkovsky Theorem

Abstract

UDC 517.9

We briefly describe some results that evolved from the Sharkovsky theorem.

References

Ll. Alsedà, D. Juher, P. Mumbrú, Periodic behavior on trees, Ergodic Theory and Dynam. Syst., 25, 1373–1400 (2005).

Ll. Alsedà, J. Llibre, M. Misiurewicz, Periodic orbits of maps of $Y$, Trans. Amer. Math. Soc., 313, 475–538 (1989).

Ll. Alsedà, J. Llibre, M. Misiurewicz, Combinatorial dynamics and entropy in dimension one, second ed., Adv. Ser. Nonlinear Dynam., 5, World Sci., Singapore (2000).

Ll. Alsedà, M. Misiurewicz, Linear orderings and the sets of periods for star maps, J. Math. Anal. and Appl., 285, 50–61 (2003).

Ll. Alsedà, J. M. Moreno, Linear orderings and the full periodicity kernel for the n-star, J. Math. Anal. and Appl., 180, 599–616 (1993).

S. Baldwin, An extension of Şarkovskií's theorem to the $n$-od}, Ergodic Theory and Dynam. Syst., 11, 249–271 (1991).

L. Block, J. Guckenheimer, M. Misiurewicz, L.-S. Young, Periodic points and topological entropy of one dimensional maps, Lecture Notes Math., 819, Springer, Berlin (1980).

L. Block, Periods of periodic points of maps of the circle which have a fixed point, Proc. Amer. Math. Soc., 82, 481–486 (1981).

A. Blokh, On some properties of graph maps: spectral decomposition, Misiurewicz conjecture and abstract sets of periods, Preprint #35, Max-Planck-Institut für Mathematik (1991).

A. Blokh, On rotation intervals for interval maps, Nonlinearity, 7, 1395–1417 (1994).

A. Blokh, Functional rotation numbers for one-dimensional maps, Trans. Amer. Math. Soc., 347, 499–514 (1995).

A. Blokh, Rotation numbers, twists and a Sharkovskii–Misiurewicz-type ordering for patterns on the interval, Ergodic Theory and Dynam. Syst., 15, 1–14 (1995).

A. Blokh, On graph-realizable sets of periods, J. Difference Equat. and Appl., 9, 343–357 (2003).

A. Blokh, M. Misiurewicz, New order for periodic orbits of interval maps, Ergodic Theory and Dynam. Syst., 17, 565–574 (1997).

A. Blokh, M. Misiurewicz, Rotation numbers for certain maps of an $n$-od, Topology and Appl., 114, 27–48 (2001).

A. Blokh, M. Misiurewicz, Forcing among patterns with no block structure, Topology Proc., 54, 125–137 (2019).

A. Blokh, M. Misiurewicz, Renormalization towers and their forcing, Trans. Amer. Math. Soc., 372, 8933–8953 (2019).

A. Blokh, O. M. Sharkovsky, Sharkovsky ordering, SpringerBriefs Math., Springer, Cham (2022).

J. Bobok, M. Kuchta, $X$-minimal patterns and a generalization of Sharkovskiv i's theorem, Fund. Math., 156, № 1, 33–66 (1998).

A. Chenciner, J.-M. Gambaudo, C. Tresser, Une remarque sur la structure des endomorphismes de degré $1$ du cercle, C. R. Acad. Sci. Paris, S'{e}r. I Math., 299, 145–148 (1984).

L. S. Efremova, Periodic orbits and the degree of a continuous map of the circle, (in Russian), Different. and Integral Equations (Gor'kií), 2, 109–115 (1978).

R. Ito, Rotation sets are closed, Math. Proc. Cambridge Phil. Soc., 89, 107–111 (1981).

M. Misiurewicz, Periodic points of maps of degree one of a circle, Ergodic Theory and Dynam. Syst., 2, 221–227 (1982).

M. Misiurewicz, Formalism for studying periodic orbits of one dimensional maps, European Conference on Iteration Theory (ECIT 87), World Sci., Singapore (1989), p.~1–7.

S. Newhouse, J. Palis, F. Takens, Bifurcations and stability of families of diffeomorphisms, Inst. Hautes '{E}tudes Sci. Publ. Math., 57, 5–71 (1983).

H. Poincaré, Sur les courbes définies par les équations différentielles, Oeuvres completes, 1, 137–158 (1952).

Scholarpedia article; http://www.scholarpedia.org/article/Rotation_theory.

A. N. Sharkovsky, Co-existence of the cycles of a continuous mapping of the line into itself} (in Russian), Ukr. Math. Zh., 16, № 1, 61–71 (1964).

A. N. Sharkovsky, Coexistence of the cycles of a continuous mapping of the line into itself, Internat. J. Bifur. and Chaos Appl. Sci. and Engrg., 5, 1263–1273 (1995).

A. N. Sharkovsky, On cycles and structure of a continuous map} (in Russian), Ukr. Math. Zh., 17, № 3, 104–111 (1965).

K. Ziemian, Rotation sets for subshifts of finite type, Fund. Math., 146, 189–201 (1995).