American Institute of Mathematical Sciences

June  2022, 42(6): 3039-3064. doi: 10.3934/dcds.2022007

A Cantor dynamical system is slow if and only if all its finite orbits are attracting

 1 AGH University of Science and Technology, Faculty of Applied Mathematics, al. Mickiewicza 30, 30-059 Kraków, Poland 2 Centre of Excellence IT4Innovations -, Institute for Research and Applications of Fuzzy Modeling, University of Ostrava, 30. dubna 22,701 03 Ostrava 1, Czech Republic

*Corresponding author: Piotr Oprocha

Received  August 2021 Revised  November 2021 Published  June 2022 Early access  January 2022

Fund Project: This work was supported by National Science Centre, Poland (NCN), grant no. 2019/35/B/ST1/02239

In this paper we completely solve the problem of when a Cantor dynamical system $(X, f)$ can be embedded in $\mathbb{R}$ with vanishing derivative. For this purpose we construct a refining sequence of marked clopen partitions of $X$ which is adapted to a dynamical system of this kind. It turns out that there is a huge class of such systems.

Citation: Silvère Gangloff, Piotr Oprocha. A Cantor dynamical system is slow if and only if all its finite orbits are attracting. Discrete and Continuous Dynamical Systems, 2022, 42 (6) : 3039-3064. doi: 10.3934/dcds.2022007
References:
 [1] L. S. Block and W. A. Coppel, Dynamics in One Dimension, Lecture Notes in Mathematics, 1513. Springer-Verlag, Berlin, 1992. doi: 10.1007/BFb0084762. [2] J. P. Boroński, J. Kupka and P. Oprocha, Edrei's conjecture revisited, Ann. Henri Poincaré, 19 (2018), 267-281.  doi: 10.1007/s00023-017-0623-9. [3] J. P. Boroński, J. Kupka and P. Oprocha, All minimal Cantor systems are slow, Bull. Lond. Math. Soc., 51 (2019), 937-944.  doi: 10.1112/blms.12275. [4] K. Ciesielski and J. Jasinski, An auto-homeomorphism of a Cantor set with derivative zero everywhere, J. Math. Anal. Appl., 434 (2016), 1267-1280.  doi: 10.1016/j.jmaa.2015.09.076. [5] M. Ciesielska and K. Ciesielski, Differentiable extension theorem: A lost proof of V. Jarník, J. Math. Anal. Appl., 454 (2017), 883-890.  doi: 10.1016/j.jmaa.2017.05.032. [6] T. Downarowicz and O. Karpel, Dynamics in dimension zero a survey, Discr. Cont. Dyn. Sys., 38 (2018), 1033-1062.  doi: 10.3934/dcds.2018044. [7] A. Edrei, On mappings which do not increase small distances, Proc. Lond. Math. Soc., 3 (1952), 272-278.  doi: 10.1112/plms/s3-2.1.272. [8] J.-M. Gambaudo and M. Martens, Algebraic topology for minimal Cantor sets, Ann. Henri Poincaré, 7 (2006), 423-446.  doi: 10.1007/s00023-005-0255-3. [9] V. Jarník, Sur l'extension du domaine de définition des fonctions d'une variable, qui laisse intacte la dérivabilité de la fonction, Bull. Internat. Acad. Sci. Boheme, (1923), 1–5. [10] K. Meydinets, Cantor aperiodic systems and Bratteli diagrams, C. R. Acad. Sci. Paris, Ser. I, 342 (2006), 43-46.  doi: 10.1016/j.crma.2005.10.024. [11] T. Shinomura, Special homeomorphisms and approximation for Cantor systems, Top. Appl., 161 (2014), 178-195.  doi: 10.1016/j.topol.2013.10.018. [12] T. Shinomura, Zero-dimensional almost 1-1 extensions of odometers from graph covering, Top. Appl., 209 (2016), 63-90.  doi: 10.1016/j.topol.2016.05.018. [13] R. F. Williams, Local contractions of compact metric sets which are not local isometries, Proc. Amer. Math. Soc., 5 (1954), 652-654.  doi: 10.1090/S0002-9939-1954-0063028-1.

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References:
 [1] L. S. Block and W. A. Coppel, Dynamics in One Dimension, Lecture Notes in Mathematics, 1513. Springer-Verlag, Berlin, 1992. doi: 10.1007/BFb0084762. [2] J. P. Boroński, J. Kupka and P. Oprocha, Edrei's conjecture revisited, Ann. Henri Poincaré, 19 (2018), 267-281.  doi: 10.1007/s00023-017-0623-9. [3] J. P. Boroński, J. Kupka and P. Oprocha, All minimal Cantor systems are slow, Bull. Lond. Math. Soc., 51 (2019), 937-944.  doi: 10.1112/blms.12275. [4] K. Ciesielski and J. Jasinski, An auto-homeomorphism of a Cantor set with derivative zero everywhere, J. Math. Anal. Appl., 434 (2016), 1267-1280.  doi: 10.1016/j.jmaa.2015.09.076. [5] M. Ciesielska and K. Ciesielski, Differentiable extension theorem: A lost proof of V. Jarník, J. Math. Anal. Appl., 454 (2017), 883-890.  doi: 10.1016/j.jmaa.2017.05.032. [6] T. Downarowicz and O. Karpel, Dynamics in dimension zero a survey, Discr. Cont. Dyn. Sys., 38 (2018), 1033-1062.  doi: 10.3934/dcds.2018044. [7] A. Edrei, On mappings which do not increase small distances, Proc. Lond. Math. Soc., 3 (1952), 272-278.  doi: 10.1112/plms/s3-2.1.272. [8] J.-M. Gambaudo and M. Martens, Algebraic topology for minimal Cantor sets, Ann. Henri Poincaré, 7 (2006), 423-446.  doi: 10.1007/s00023-005-0255-3. [9] V. Jarník, Sur l'extension du domaine de définition des fonctions d'une variable, qui laisse intacte la dérivabilité de la fonction, Bull. Internat. Acad. Sci. Boheme, (1923), 1–5. [10] K. Meydinets, Cantor aperiodic systems and Bratteli diagrams, C. R. Acad. Sci. Paris, Ser. I, 342 (2006), 43-46.  doi: 10.1016/j.crma.2005.10.024. [11] T. Shinomura, Special homeomorphisms and approximation for Cantor systems, Top. Appl., 161 (2014), 178-195.  doi: 10.1016/j.topol.2013.10.018. [12] T. Shinomura, Zero-dimensional almost 1-1 extensions of odometers from graph covering, Top. Appl., 209 (2016), 63-90.  doi: 10.1016/j.topol.2016.05.018. [13] R. F. Williams, Local contractions of compact metric sets which are not local isometries, Proc. Amer. Math. Soc., 5 (1954), 652-654.  doi: 10.1090/S0002-9939-1954-0063028-1.
In this representation, the two periodic orbits of the systems are not distinguished in the first graph
Illustration of the graph $G(\mathcal{U})$ for a supercyclical partition $\mathcal{U}$ of $(X, f)$. The dashed regions correspond to the attracted part of the partition, the remainder corresponds to the supercyclical part
Removing the divergent vertices in the attracted part
Two finite directed graphs $G = (V, E)$ (left) and $G' = (V', E')$ (right) and $\pi: V\rightarrow V'$ morphism, sending vertices of $G$ to the one of the same color in $G'$. The graph $G'$ can correspond to a well-marked partition $(\mathcal{V}, \tau, \chi)$. However it is not possible to find a partition whose graph is $G$ and which would be well-marked relatively to $(\mathcal{V}, \tau, \chi)$. Indeed, whatever the way we mark the red vertices, there will be at least one circuit left with no marker or no potential
Illustration on an example of the definition of the graphs $\mathcal{I}(G, \chi)$ and $\mathcal{A}(G, \chi)$ for the graph $G = G(\mathcal{U})$ and $\chi: \mathcal{U} \rightarrow \{0, \downarrow, \uparrow, *\}$ where $(\mathcal{U}, \chi)$ is a well-marked partition; the function $\chi$ is partially represented (for simplicity): only markers and vertices with $\chi(u) = 0$ (the ones in dashed regions) are represented
Illustration of the definition of $\iota_n$ on preimages of some vertex in $G_{n-1}$ when this vertex is not in a circuit corresponding to a finite orbit
Illustration of the definition of $\iota_n(v)$ for $v$ a preimage by $\pi_{n-1}$ of $w$ which is in a circuit of the attracted part of $\mathcal{U}_{n-1}$
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