Abstract
We prove an upper bound for the polynomial entropy of continuous, piecewise monotone maps of the interval, according to the number of intervals of monotonicity of its iterates. We give examples that show that this inequality is sharp. As a direct consequence of this inequality, the polynomial entropy of monotone, continuous, interval maps is always less than or equal to one. We give examples where we can also obtain lower bounds. We also prove analogous inequality in terms of total variations of the iterates of these interval maps. Also, this inequality is sharp.
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References
Artigue, A., Carrasco-Oliveira, D., Monteverde, I.: Polynomial entropy and expansivity. Acta Math. Hung. 152, 140–149 (2017)
Baillif, M.: A polynomial bound for the lap number. Qual. Theory Dyn. Syst. 3, 325–329 (2002)
Bowen, R., Franks, J.: The periodic points of maps of the disk and the interval. Topology 15, 337–342 (1976)
Brucks, K.M., Bruin, H.: Topics from One-Dimensional Dynamics. Cambridge University Press, Cambridge (2004)
Kozlovski, O., Shen, W., van Strien, S.: Density of hyperbolicity in dimension one. Ann. Math. 166, 145–182 (2007)
Labrousse, C.: Polynomial entropy for the circle homeomorphisms and for \(C^1\) nonvanishing vector fields on \({\mathbb{T}}^2\). arXiv: 1311.02131v1
Marco, J.-P.: Polynomial entropies and integrable Hamiltonian systems. Regul. Chaotic Dyn. 18, 623–655 (2013)
Milnor, J., Thurston, W.: On iterated maps of the interval. Lect. Notes Math. 1342, 465–563 (2006)
Misiurewicz, M.: Structure of mappings of an interval with zero entropy. Publ. Math. IHES Paris 53, 5–16 (1981)
Misiurewicz, M., Szlenk, W.: Entropy of piecewise monotone mappings. Stud. Math. LXVII, 45–63 (1980)
Nitecki, Z.: Topological dynamics on the interval. In: Katok, A. (ed.) Ergodic Theory and Dynamical Systems II, pp. 1–73. Springer, New York (1982)
Rothschild, J.: On the computation of topological entropy. Thesis, CUNI (1971)
Ruette, S.: Chaos on the Interval. University Lecture Series, vol. 67. American Mathematical Society, Providence (2017)
Walters, P.: An Introduction to Ergodic Theory. Springer, Berlin (1982)
Young, L.-S.: On the prevalence of horseshoes. Trans. Am. Math. Soc. 263, 75–88 (1981)
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We thank the anonymous referee(s) for the careful reading and most invaluable suggestions.
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Gomes, J.B., Carneiro, M.J.D. Polynomial Entropy for Interval Maps and Lap Number. Qual. Theory Dyn. Syst. 20, 21 (2021). https://doi.org/10.1007/s12346-021-00456-y
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DOI: https://doi.org/10.1007/s12346-021-00456-y