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The period set of a map from the Cantor set to itself
1. | Mathematics Department, Brigham Young University, Provo, UT, 84602, United States |
2. | Mathematics Department, Southern Utah University, Cedar City, UT, 84720, United States |
3. | Institute of Mathematics, University of Gdańsk ul. Wita Stwosza 57, PL-80952 Gdańsk, Poland |
References:
[1] |
Ll. Alsedà, D. Juher and P. Mumbrú, Sets of periods for piecewise monotone tree maps, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 13 (2003), 311-341.
doi: 10.1142/S021812740300656X. |
[2] |
Lluís Alsedà, Jaume Llibre and Michał Misiurewicz, Periodic orbits of maps of $Y$, Trans. Amer. Math. Soc., 313 (1989), 475-538.
doi: 10.2307/2001417. |
[3] |
Lluís Alsedà i Soler, Periodic points of continuous mappings of the circle, Publ. Sec. Mat. Univ. Autònoma Barcelona, 24 (1981), 5-71. |
[4] |
Stewart Baldwin, An extension of Šarkovskiĭ's theorem to the $n-od$, Ergodic Theory Dynam. Systems, 11 (1991), 249-271.
doi: 10.1017/S0143385700006131. |
[5] |
Stewart Baldwin, Versions of Sharkovskiĭ's theorem on trees and dendrites, Topology Proc., 18 (1993), 19-31. |
[6] |
Louis Block, John Guckenheimer, Michał Misiurewicz and Lai Sang Young, Periodic points and topological entropy of one-dimensional maps, in "Global Theory of Dynamical Systems (Proc. Internat. Conf., Northwestern Univ., Evanston, Ill., 1979)" 819 of Lecture Notes in Math., 18-34. Springer, Berlin, (1980). |
[7] |
A. I. Demin, Coexistence of periodic, almost periodic and recurrent points of transformations of $n-od$, Vestnik Moskov. Univ. Ser. I Mat. Mekh., 3 (1996), 84-87. |
[8] |
Patrick Gallagher, Approximation by reduced fractions, J. Math. Soc. Japan, 13 (1961), 342-345. |
[9] |
Christian Gillot and Jaume Llibre, Periods for maps of the figure-eight space, in "Thirty Years After Sharkovskiĭ's Theorem: New Perspectives (Murcia, 1994)" 8 of World Sci. Ser. Nonlinear Sci. Ser. B Spec. Theme Issues Proc., 95-106. World Sci. Publ., River Edge, NJ, (1995). Reprint of the paper reviewed in MR1361924 (97d:58161). |
[10] |
W. T. Ingram, Periodic points for homeomorphisms of hereditarily decomposable chainable continua, Proc. Amer. Math. Soc., 107 (1989), 549-553.
doi: 10.2307/2047846. |
[11] |
Mark H. Meilstrup, "Wild Low-Dimensional Topology and Dynamics," Ph.D thesis, Brigham Young University, 2010. |
[12] |
Michał Misiurewicz, Periodic points of maps of degree one of a circle, Ergodic Theory Dynamical Systems, 2 (1982), 221-227. |
[13] |
E. Mochko, V. V. Nekrashevich and V. I. Sushchanskiĭ, Dynamics of triangular transformations of sequences over finite alphabets, Mat. Zametki, 73 (2003), 466-468. Translation in Math. Notes, 73 (2003), 436-439.
doi: 10.1023/A:1023234532265. |
[14] |
T. Pezda, Polynomial cycles in certain local domains, Acta Arith., 66 (1994), 11-22. |
[15] |
A. N. Sharkovskiĭ, Coexistence of cycles of a continuous map of the line into itself, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 5 (1995), 1263-1273. Translated from the Russian Ukrain. Mat. Zh., 16 (1964), 61-71 by J. Tolosa.
doi: 10.1142/S0218127495000934. |
[16] |
H. W. Siegberg, Chaotic mappings on $S^1$, periods one, two, three imply chaos on $S^1$, in "Numerical Solution of Nonlinear Equations (Bremen, 1980)" 878 of Lecture Notes in Math., 351-370. Springer, Berlin, (1981). |
[17] |
V. I. Sushchanski, E. Moćko and V. V. Nekrashevych, Cycles of distance-decreasing mappings in the ring of $n$-adic integers, Colloq. Math., 105 (2006), 197-205.
doi: 10.4064/cm105-2-3. |
show all references
References:
[1] |
Ll. Alsedà, D. Juher and P. Mumbrú, Sets of periods for piecewise monotone tree maps, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 13 (2003), 311-341.
doi: 10.1142/S021812740300656X. |
[2] |
Lluís Alsedà, Jaume Llibre and Michał Misiurewicz, Periodic orbits of maps of $Y$, Trans. Amer. Math. Soc., 313 (1989), 475-538.
doi: 10.2307/2001417. |
[3] |
Lluís Alsedà i Soler, Periodic points of continuous mappings of the circle, Publ. Sec. Mat. Univ. Autònoma Barcelona, 24 (1981), 5-71. |
[4] |
Stewart Baldwin, An extension of Šarkovskiĭ's theorem to the $n-od$, Ergodic Theory Dynam. Systems, 11 (1991), 249-271.
doi: 10.1017/S0143385700006131. |
[5] |
Stewart Baldwin, Versions of Sharkovskiĭ's theorem on trees and dendrites, Topology Proc., 18 (1993), 19-31. |
[6] |
Louis Block, John Guckenheimer, Michał Misiurewicz and Lai Sang Young, Periodic points and topological entropy of one-dimensional maps, in "Global Theory of Dynamical Systems (Proc. Internat. Conf., Northwestern Univ., Evanston, Ill., 1979)" 819 of Lecture Notes in Math., 18-34. Springer, Berlin, (1980). |
[7] |
A. I. Demin, Coexistence of periodic, almost periodic and recurrent points of transformations of $n-od$, Vestnik Moskov. Univ. Ser. I Mat. Mekh., 3 (1996), 84-87. |
[8] |
Patrick Gallagher, Approximation by reduced fractions, J. Math. Soc. Japan, 13 (1961), 342-345. |
[9] |
Christian Gillot and Jaume Llibre, Periods for maps of the figure-eight space, in "Thirty Years After Sharkovskiĭ's Theorem: New Perspectives (Murcia, 1994)" 8 of World Sci. Ser. Nonlinear Sci. Ser. B Spec. Theme Issues Proc., 95-106. World Sci. Publ., River Edge, NJ, (1995). Reprint of the paper reviewed in MR1361924 (97d:58161). |
[10] |
W. T. Ingram, Periodic points for homeomorphisms of hereditarily decomposable chainable continua, Proc. Amer. Math. Soc., 107 (1989), 549-553.
doi: 10.2307/2047846. |
[11] |
Mark H. Meilstrup, "Wild Low-Dimensional Topology and Dynamics," Ph.D thesis, Brigham Young University, 2010. |
[12] |
Michał Misiurewicz, Periodic points of maps of degree one of a circle, Ergodic Theory Dynamical Systems, 2 (1982), 221-227. |
[13] |
E. Mochko, V. V. Nekrashevich and V. I. Sushchanskiĭ, Dynamics of triangular transformations of sequences over finite alphabets, Mat. Zametki, 73 (2003), 466-468. Translation in Math. Notes, 73 (2003), 436-439.
doi: 10.1023/A:1023234532265. |
[14] |
T. Pezda, Polynomial cycles in certain local domains, Acta Arith., 66 (1994), 11-22. |
[15] |
A. N. Sharkovskiĭ, Coexistence of cycles of a continuous map of the line into itself, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 5 (1995), 1263-1273. Translated from the Russian Ukrain. Mat. Zh., 16 (1964), 61-71 by J. Tolosa.
doi: 10.1142/S0218127495000934. |
[16] |
H. W. Siegberg, Chaotic mappings on $S^1$, periods one, two, three imply chaos on $S^1$, in "Numerical Solution of Nonlinear Equations (Bremen, 1980)" 878 of Lecture Notes in Math., 351-370. Springer, Berlin, (1981). |
[17] |
V. I. Sushchanski, E. Moćko and V. V. Nekrashevych, Cycles of distance-decreasing mappings in the ring of $n$-adic integers, Colloq. Math., 105 (2006), 197-205.
doi: 10.4064/cm105-2-3. |
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