American Institute of Mathematical Sciences

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

 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 $B$.
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