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

Анотація

УДК 517.9

Еволюція теореми Шарковського

Коротко описано деякі результати, що розвинулися з теореми Шарковського.

Посилання

Ll. Alsedà, D. Juher, P. Mumbrú, Periodic behavior on trees, Ergodic Theory and Dynam. Syst., 25, 1373–1400 (2005). DOI: https://doi.org/10.1017/S0143385704000896

Ll. Alsedà, J. Llibre, M. Misiurewicz, Periodic orbits of maps of $Y$, Trans. Amer. Math. Soc., 313, 475–538 (1989). DOI: https://doi.org/10.1090/S0002-9947-1989-0958882-0

Ll. Alsedà, J. Llibre, M. Misiurewicz, Combinatorial dynamics and entropy in dimension one, second ed., Adv. Ser. Nonlinear Dynam., 5, World Sci., Singapore (2000). DOI: https://doi.org/10.1142/4205

Ll. Alsedà, M. Misiurewicz, Linear orderings and the sets of periods for star maps, J. Math. Anal. and Appl., 285, 50–61 (2003). DOI: https://doi.org/10.1016/S0022-247X(02)00659-5

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). DOI: https://doi.org/10.1006/jmaa.1993.1419

S. Baldwin, An extension of Şarkovskií's theorem to the $n$-od}, Ergodic Theory and Dynam. Syst., 11, 249–271 (1991). DOI: https://doi.org/10.1017/S0143385700006131

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). DOI: https://doi.org/10.1007/BFb0086977

L. Block, Periods of periodic points of maps of the circle which have a fixed point, Proc. Amer. Math. Soc., 82, 481–486 (1981). DOI: https://doi.org/10.1090/S0002-9939-1981-0612745-7

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). DOI: https://doi.org/10.1088/0951-7715/7/5/008

A. Blokh, Functional rotation numbers for one-dimensional maps, Trans. Amer. Math. Soc., 347, 499–514 (1995). DOI: https://doi.org/10.2307/2154898

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). DOI: https://doi.org/10.1017/S014338570000821X

A. Blokh, On graph-realizable sets of periods, J. Difference Equat. and Appl., 9, 343–357 (2003). DOI: https://doi.org/10.1080/1023619021000047770

A. Blokh, M. Misiurewicz, New order for periodic orbits of interval maps, Ergodic Theory and Dynam. Syst., 17, 565–574 (1997). DOI: https://doi.org/10.1017/S0143385797084927

A. Blokh, M. Misiurewicz, Rotation numbers for certain maps of an $n$-od, Topology and Appl., 114, 27–48 (2001). DOI: https://doi.org/10.1016/S0166-8641(00)00033-X

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). DOI: https://doi.org/10.1090/tran/7928

A. Blokh, O. M. Sharkovsky, Sharkovsky ordering, SpringerBriefs Math., Springer, Cham (2022). DOI: https://doi.org/10.1007/978-3-030-99125-8

J. Bobok, M. Kuchta, $X$-minimal patterns and a generalization of Sharkovskiv i's theorem, Fund. Math., 156, № 1, 33–66 (1998). DOI: https://doi.org/10.4064/fm-156-1-33-66

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). DOI: https://doi.org/10.1017/S0305004100057984

M. Misiurewicz, Periodic points of maps of degree one of a circle, Ergodic Theory and Dynam. Syst., 2, 221–227 (1982). DOI: https://doi.org/10.1017/S014338570000153X

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). DOI: https://doi.org/10.1007/BF02698773

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). DOI: https://doi.org/10.1142/S0218127495000934

A. N. Sharkovsky, On cycles and structure of a continuous map} (in Russian), Ukr. Math. Zh., 17, № 3, 104–111 (1965). DOI: https://doi.org/10.1007/BF02527365

K. Ziemian, Rotation sets for subshifts of finite type, Fund. Math., 146, 189–201 (1995). DOI: https://doi.org/10.4064/fm-146-2-189-201