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Abstract We analyze the q -deformed logistic map, where the q -deformation follows the scheme inspired in the Tsallis q -exponential function. We compute the topological entropy of the dynamical system, obtaining the parametric region in which the topological entropy is positive and hence the region in which chaos in the sense of Li and Yorke exists. In addition, it is shown the existence of the so-called Parrondo's paradox where two simple maps are combined to give a complicated dynamical behavior. Highlights • The q -deformed logistic map is analyzed by computing its topological entropy and Lyapunov exponents. • It is shown that it is useful to see the q -deformation as the composition of two maps. • The existence of Parrondo's paradox "simple+simple gives complex" is shown. [ABSTRACT FROM AUTHOR]