American Institute of Mathematical Sciences

Advanced
July  2013, 33(7): 2667-2679. doi: 10.3934/dcds.2013.33.2667

The period set of a map from the Cantor set to itself

James W. Cannon 1, , Mark H. Meilstrup 2, and  Andreas Zastrow 3,

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

Received  March 2012 Revised  August 2012 Published  January 2013

In this paper we consider all possible period sets $P(f)$ for self-maps of the Cantor set, $f:C\to C$. We prove that the possible period sets are completely unrestricted provided that, in addition, one allows points that are not periodic. However, if every point is periodic, we show that a surprising finiteness condition is imposed on $P(f)$: namely, there is a finite subset $B$ of $P(f)$ such that every element of $P(f)$ is divisible by at least one element of $B$.
Keywords: periodic points, aperiodic points, period set, backtracking., Cantor set, Sharkovsky's Theorem.
Mathematics Subject Classification: 37E15, 54H20, 37B1.
Citation: James W. Cannon, Mark H. Meilstrup, Andreas Zastrow. The period set of a map from the Cantor set to itself. Discrete & Continuous Dynamical Systems - A, 2013, 33 (7) : 2667-2679. doi: 10.3934/dcds.2013.33.2667
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.  doi: 10.1142/S021812740300656X.  Google Scholar

[2]

Lluís Alsedà, Jaume Llibre and Michał Misiurewicz, Periodic orbits of maps of $Y$,, Trans. Amer. Math. Soc., 313 (1989), 475.  doi: 10.2307/2001417.  Google Scholar

[3]

Lluís Alsedà i Soler, Periodic points of continuous mappings of the circle,, Publ. Sec. Mat. Univ. Autònoma Barcelona, 24 (1981), 5.   Google Scholar

[4]

Stewart Baldwin, An extension of Šarkovskiĭ's theorem to the $n-od$,, Ergodic Theory Dynam. Systems, 11 (1991), 249.  doi: 10.1017/S0143385700006131.  Google Scholar

[5]

Stewart Baldwin, Versions of Sharkovskiĭ's theorem on trees and dendrites,, Topology Proc., 18 (1993), 19.   Google Scholar

[6]

Louis Block, John Guckenheimer, Michał Misiurewicz and Lai Sang Young, Periodic points and topological entropy of one-dimensional maps,, in, 819 (1980), 18.   Google Scholar

[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.   Google Scholar

[8]

Patrick Gallagher, Approximation by reduced fractions,, J. Math. Soc. Japan, 13 (1961), 342.   Google Scholar

[9]

Christian Gillot and Jaume Llibre, Periods for maps of the figure-eight space,, in, 8 (1995), 95.   Google Scholar

[10]

W. T. Ingram, Periodic points for homeomorphisms of hereditarily decomposable chainable continua,, Proc. Amer. Math. Soc., 107 (1989), 549.  doi: 10.2307/2047846.  Google Scholar

[11]

Mark H. Meilstrup, "Wild Low-Dimensional Topology and Dynamics,", Ph.D thesis, (2010).   Google Scholar

[12]

Michał Misiurewicz, Periodic points of maps of degree one of a circle,, Ergodic Theory Dynamical Systems, 2 (1982), 221.   Google Scholar

[13]

E. Mochko, V. V. Nekrashevich and V. I. Sushchanskiĭ, Dynamics of triangular transformations of sequences over finite alphabets,, Mat. Zametki, 73 (2003), 466.  doi: 10.1023/A:1023234532265.  Google Scholar

[14]

T. Pezda, Polynomial cycles in certain local domains,, Acta Arith., 66 (1994), 11.   Google Scholar

[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.  doi: 10.1142/S0218127495000934.  Google Scholar

[16]

H. W. Siegberg, Chaotic mappings on $S^1$, periods one, two, three imply chaos on $S^1$,, in, 878 (1981), 351.   Google Scholar

[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.  doi: 10.4064/cm105-2-3.  Google Scholar

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.  doi: 10.1142/S021812740300656X.  Google Scholar

[2]

Lluís Alsedà, Jaume Llibre and Michał Misiurewicz, Periodic orbits of maps of $Y$,, Trans. Amer. Math. Soc., 313 (1989), 475.  doi: 10.2307/2001417.  Google Scholar

[3]

Lluís Alsedà i Soler, Periodic points of continuous mappings of the circle,, Publ. Sec. Mat. Univ. Autònoma Barcelona, 24 (1981), 5.   Google Scholar

[4]

Stewart Baldwin, An extension of Šarkovskiĭ's theorem to the $n-od$,, Ergodic Theory Dynam. Systems, 11 (1991), 249.  doi: 10.1017/S0143385700006131.  Google Scholar

[5]

Stewart Baldwin, Versions of Sharkovskiĭ's theorem on trees and dendrites,, Topology Proc., 18 (1993), 19.   Google Scholar

[6]

Louis Block, John Guckenheimer, Michał Misiurewicz and Lai Sang Young, Periodic points and topological entropy of one-dimensional maps,, in, 819 (1980), 18.   Google Scholar

[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.   Google Scholar

[8]

Patrick Gallagher, Approximation by reduced fractions,, J. Math. Soc. Japan, 13 (1961), 342.   Google Scholar

[9]

Christian Gillot and Jaume Llibre, Periods for maps of the figure-eight space,, in, 8 (1995), 95.   Google Scholar

[10]

W. T. Ingram, Periodic points for homeomorphisms of hereditarily decomposable chainable continua,, Proc. Amer. Math. Soc., 107 (1989), 549.  doi: 10.2307/2047846.  Google Scholar

[11]

Mark H. Meilstrup, "Wild Low-Dimensional Topology and Dynamics,", Ph.D thesis, (2010).   Google Scholar

[12]

Michał Misiurewicz, Periodic points of maps of degree one of a circle,, Ergodic Theory Dynamical Systems, 2 (1982), 221.   Google Scholar

[13]

E. Mochko, V. V. Nekrashevich and V. I. Sushchanskiĭ, Dynamics of triangular transformations of sequences over finite alphabets,, Mat. Zametki, 73 (2003), 466.  doi: 10.1023/A:1023234532265.  Google Scholar

[14]

T. Pezda, Polynomial cycles in certain local domains,, Acta Arith., 66 (1994), 11.   Google Scholar

[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.  doi: 10.1142/S0218127495000934.  Google Scholar

[16]

H. W. Siegberg, Chaotic mappings on $S^1$, periods one, two, three imply chaos on $S^1$,, in, 878 (1981), 351.   Google Scholar

[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.  doi: 10.4064/cm105-2-3.  Google Scholar

[1]

Fabian Ziltener. Note on coisotropic Floer homology and leafwise fixed points. Electronic Research Archive, , () : -. doi: 10.3934/era.2021001
[2]

Bing Yu, Lei Zhang. Global optimization-based dimer method for finding saddle points. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 741-753. doi: 10.3934/dcdsb.2020139
[3]

Huu-Quang Nguyen, Ya-Chi Chu, Ruey-Lin Sheu. On the convexity for the range set of two quadratic functions. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020169
[4]

Sumit Kumar Debnath, Pantelimon Stǎnicǎ, Nibedita Kundu, Tanmay Choudhury. Secure and efficient multiparty private set intersection cardinality. Advances in Mathematics of Communications, 2021, 15 (2) : 365-386. doi: 10.3934/amc.2020071
[5]

Yasmine Cherfaoui, Mustapha Moulaï. Biobjective optimization over the efficient set of multiobjective integer programming problem. Journal of Industrial & Management Optimization, 2021, 17 (1) : 117-131. doi: 10.3934/jimo.2019102
[6]

Shasha Hu, Yihong Xu, Yuhan Zhang. Second-Order characterizations for set-valued equilibrium problems with variable ordering structures. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020164
[7]

Wenbin Li, Jianliang Qian. Simultaneously recovering both domain and varying density in inverse gravimetry by efficient level-set methods. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2020073
[8]

Ali Mahmoodirad, Harish Garg, Sadegh Niroomand. Solving fuzzy linear fractional set covering problem by a goal programming based solution approach. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020162
[9]

Lingfeng Li, Shousheng Luo, Xue-Cheng Tai, Jiang Yang. A new variational approach based on level-set function for convex hull problem with outliers. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2020070
[10]

Darko Dimitrov, Hosam Abdo. Tight independent set neighborhood union condition for fractional critical deleted graphs and ID deleted graphs. Discrete & Continuous Dynamical Systems - S, 2019, 12 (4&5) : 711-721. doi: 10.3934/dcdss.2019045
[11]

Jingjing Wang, Zaiyun Peng, Zhi Lin, Daqiong Zhou. On the stability of solutions for the generalized vector quasi-equilibrium problems via free-disposal set. Journal of Industrial & Management Optimization, 2021, 17 (2) : 869-887. doi: 10.3934/jimo.2020002
[12]

Peter Frolkovič, Viera Kleinová. A new numerical method for level set motion in normal direction used in optical flow estimation. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 851-863. doi: 10.3934/dcdss.2020347
[13]

Tetsuya Ishiwata, Takeshi Ohtsuka. Numerical analysis of an ODE and a level set methods for evolving spirals by crystalline eikonal-curvature flow. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 893-907. doi: 10.3934/dcdss.2020390
[14]

Chao Wang, Qihuai Liu, Zhiguo Wang. Periodic bouncing solutions for Hill's type sub-linear oscillators with obstacles. Communications on Pure & Applied Analysis, 2021, 20 (1) : 281-300. doi: 10.3934/cpaa.2020266
[15]

Yuanfen Xiao. Mean Li-Yorke chaotic set along polynomial sequence with full Hausdorff dimension for $ \beta $-transformation. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 525-536. doi: 10.3934/dcds.2020267
[16]

Duy Phan. Approximate controllability for Navier–Stokes equations in $ \rm3D $ cylinders under Lions boundary conditions by an explicit saturating set. Evolution Equations & Control Theory, 2021, 10 (1) : 199-227. doi: 10.3934/eect.2020062
[17]

Peng Luo. Comparison theorem for diagonally quadratic BSDEs. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020374
[18]

Yukihiko Nakata. Existence of a period two solution of a delay differential equation. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 1103-1110. doi: 10.3934/dcdss.2020392
[19]

Hongguang Ma, Xiang Li. Multi-period hazardous waste collection planning with consideration of risk stability. Journal of Industrial & Management Optimization, 2021, 17 (1) : 393-408. doi: 10.3934/jimo.2019117
[20]

Attila Dénes, Gergely Röst. Single species population dynamics in seasonal environment with short reproduction period. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020288

2019 Impact Factor: 1.338

Tools

Metrics

  • PDF downloads (41)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences