Remark on Transitivity for piecewise increassing maps on R


  • Luis Bladismir Ruiz Leal Instituto de Ciencias Básicas, Universidad Técnica de Manabí, Av. Urbina y Che Guevara, Portoviejo - Ecuador.
  • Ambrosio Tineo Instituto de Ciencias Básicas, Universidad Técnica de Manabí, Av. Urbina y Che Guevara, Portoviejo - Ecuador.
  • Abdul Lugo Instituto Superior de Formación Docente Salomé Ureña, Recinto Félix Evaristo Mejía. Santo Domingo - República Dominicana.



Transitivity maps, piecewise increassing maps, vertical asymptote


In this work a sufficient condition is shown to obtain transitivity in families of piecewise increassing maps with an inevitable discontinuity in x=0. Specifically, it is shown that the characteristics of a large class of transformations of the real line with a discontinuity in x=0 to be transitive (exhibits a dense orbit), they are the following: f has no fixed points, f has a vertical asymptote at x=0 and the preimage of zero is different from empty. In particular, the famous Boole transformation together with some of its parameterizations they exhibit these characteristics. As a particular case, for the family to a parameter of hyperbolas its dynamic behavior is explicitly determined according to the values of the parameter p > 0.


Aaronson J. The eigenvalues of non-singular transformations, Israel J. Math. 1983; 45:297-312.

Aaronson J. An Introduction to Infinite Ergodic Theory. Mathematical Surveys and Monographs of the Amer. Math. Soc., Vol 50. 1997.

Adler R, Weiss B. The ergodic infinite measure preserving transformation of Boole, Israel J. Math. 1973; 16:263-278.

Bayless RL. Ergodic Properties of Rational Functions that Preserve Lebesgue Measure on R. Real Analysis Exchange. 2018; Vol. 43(1):137-153.

Blackmore D, Golenia J, Prykarpatsky AK, et al. Invariant measures for discrete dynamical systems and ergodic properties of generalized Boole-type transformations. Ukr Math J. 2013; 65:47.

Bonanno C, Giulietti P, Lenci M. Global-local mixing for the Boole map, Chaos, Solitons & Fractals. 2018; 111:55-61.

Leal B, Mata G, Muñoz S. Families of Transitive Maps on R With Horizontal Asymptotes. Rev. de la Unión Matemática Argentina (REVUMA). 2018; 58(2):375-387.

Letac G. Which Functions Preserve Cauchy Laws?. Proc. Amer. Math. Soc.1977; 67(2):277–286.

Li T, Schweiger F. The Generalized Boole’s Transformation is ergodic, Manuscripta Math. 1971; 25:161-167.

Muñoz S. Robust transitivity and ergodicity of transformations of the real line and the real plane[PhD Thesis], IMPA; 2006. Available at

Muñoz S. Robust transitivity of maps of the real line. Discrete and Continuous Dynamical Systems. Series A. 2015; 35(3):1163-1177. avilable at

Neuwirth JH. Ergodicity of some mapping of the circle and the line. Israel J. Math. 1978; 31:359-367.

Prykarpatsky AK, Feldman J. On the ergodic and spectral properties of generalied Boole transformations. I. Miskolc Math. Notes. 2006; 7(1):91-99.

Umento K, Okubo K. Exact Lyapunov exponents of the generalized Boole transformations. Prog. Theor. Exp. Phys. 2016; 021A01.



How to Cite

Ruiz Leal, L. B., Tineo, A., & Lugo, A. (2022). Remark on Transitivity for piecewise increassing maps on R. Selecciones Matemáticas, 9(01), 145 - 149.