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Mixing properties of erasing interval maps

Part of: Discrete mathematics in relation to computer science Ergodic theory Measure-theoretic ergodic theory

Published online by Cambridge University Press:  06 March 2023

DARIO CORONA Open the ORCID record for DARIO CORONA [Opens in a new window]  and
ALESSANDRO DELLA CORTE Open the ORCID record for ALESSANDRO DELLA CORTE [Opens in a new window]
DARIO CORONA
Affiliation:
Mathematics Division, School of Sciences and Technology, University of Camerino, Camerino 62032, Italy (e-mail: dario.corona@unicam.it)
ALESSANDRO DELLA CORTE*
Affiliation:
Mathematics Division, School of Sciences and Technology, University of Camerino, Camerino 62032, Italy (e-mail: dario.corona@unicam.it)
*
e-mail: alessandro.dellacorte@unicam.it
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Abstract

We study the measurable dynamical properties of the interval map generated by the model-case erasing substitution $\rho $, defined by

$$ \begin{align*} \rho(00)=\text{empty word},\quad \rho(01)=1,\quad \rho(10)=0,\quad \rho(11)=01. \end{align*} $$
We prove that, although the map is singular, its square preserves the Lebesgue measure and is strongly mixing, thus ergodic, with respect to it. We discuss the extension of the results to more general erasing maps.

Keywords

mixingsubstitutionscombinatorics on words

MSC classification

Primary: 37A25: Ergodicity, mixing, rates of mixing 28D05: Measure-preserving transformations 37A05: Measure-preserving transformations
Secondary: 68R15: Combinatorics on words
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 44 , Issue 2 , February 2024 , pp. 408 - 431
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Copyright
© The Author(s), 2023. Published by Cambridge University Press

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References

Balibrea, F., Smıtal, J. and Štefánková, M.. The three versions of distributional chaos. Chaos Solitons Fractals 23(5) (2005), 1581–1583.Google Scholar
Bobok, J. and Troubetzkoy, S.. Typical properties of interval maps preserving the Lebesgue measure. Nonlinearity 33(12) (2020), 6461.CrossRefGoogle Scholar
Bowen, R.. Entropy for group endomorphisms and homogeneous spaces. Trans. Amer. Math. Soc. 153 (1971), 401–414.CrossRefGoogle Scholar
Čiklová, M.. Dynamical systems generated by functions with connected ${G}_{\delta }$ graphs. Real Anal. Exchange 30(2) (2005), 617–638.CrossRefGoogle Scholar
Della Corte, A. and Farotti, M.. Dynamics of interval maps generated by erasing substitutions. J. Differential Equations 323 (2022), 86–112.10.1016/j.jde.2022.03.036CrossRefGoogle Scholar
Della Corte, A., Isola, S. and Piergallini, R.. The simplest erasing substitution. Nonlinearity 35 (2022), 4317–4356.CrossRefGoogle Scholar
Dinaburg, E. I.. A correlation between topological entropy and metric entropy. Dokl. Akad. Nauk 190(1) (1970), 19–22.Google Scholar
Downarowicz, T.. Entropy structure. J. Anal. Math. 96(1) (2005), 57–116.CrossRefGoogle Scholar
Downarowicz, T.. Entropy in Dynamical Systems (New Mathematical Monographs, 18). Cambridge University Press, Cambridge, 2011.10.1017/CBO9780511976155CrossRefGoogle Scholar
Durand, F.. Cobham’s theorem for substitutions. J. Eur. Math. Soc. (JEMS) 13(6) (2011), 1799–1814.10.4171/jems/294CrossRefGoogle Scholar
Durand, F. and Rigo, M.. Syndeticity and independent substitutions. Adv. Appl. Math. 42(1) (2009), 1–22.CrossRefGoogle Scholar
Einsiedler, M. and Ward, T.. Ergodic Theory. Springer, London, 2013, pp. 4–5.Google Scholar
Fogg, N. P., Berthé, V., Ferenczi, S., Mauduit, C. and Siegel, A. (Eds.). Substitutions in Dynamics, Arithmetics and Combinatorics. Springer, Berlin, 2002.CrossRefGoogle Scholar
Gentili, S.. Semi-algebra and algebra of sets. Measure, Integration and a Primer on Probability Theory. (UNITEXT, 125). Springer, Cham, 2020, pp. 121–129.10.1007/978-3-030-54940-4_6CrossRefGoogle Scholar
Loranty, A., Pawlak, R. J. and Terepeta, M.. On almost continuous functions and peculiar points. Eur. J. Math. 5(1) (2019), 106–115.10.1007/s40879-018-0264-7CrossRefGoogle Scholar
Pawlak, R.. On the entropy of Darboux functions. Colloq. Math. 2 (2009), 116.Google Scholar
Petersen, K.. Ergodic Theory (Cambridge Studies in Advanced Mathematics, 2). Cambridge University Press, Cambridge, 1983.CrossRefGoogle Scholar
Queffelec, M.. Old and new results on normality. IMS Lecture Notes Monogr. Ser. 48 (2006), 225–236.Google Scholar
Queffélec, M.. Substitution Dynamical Systems–Spectral Analysis (Lecture Notes in Mathematics, 1294). Springer, Berlin, 2010.CrossRefGoogle Scholar
Reidenbach, D. and Schneider, J. C.. Restricted ambiguity of erasing morphisms. Theoret. Comput. Sci. 412(29) (2011), 3510–3523.CrossRefGoogle Scholar
Steele, T. H.. Dynamics of Baire-1 functions on the interval. Eur. J. Math. 5(1) (2019), 138–149.10.1007/s40879-018-0234-0CrossRefGoogle Scholar
Steele, T. H.. Dynamics of Baire-2 functions on the interval. Topology Appl. 265 (2019), 106821.CrossRefGoogle Scholar