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The purpose of this paper is to explore new mathematical results to advance the understanding of the picture of a chaotic unimodal map.

Ever since Poicare, deterministic chaos is ultimately connected with exponential divergence of nearby trajectories, unpredictability and erratic behaviour. Here, the authors propose an alternative approach in terms of complexity theory and transcendence.

In this paper, the authors were able to reproduce previous results easily, due to new theorems.

The paper updates previous results and proposes a more complete understanding of the phenomenon of deterministic chaos, also making possible connections with number theory, combinatorics and possibly quantum mechanics, as in quantum mechanics there does not exist the notion on nearby trajectories.