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The mean-field limit of the Lieb-Liniger model
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 |
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.
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. |
show all references
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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