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Fig. 2 | Bollettino dell'Unione Matematica Italiana

Fig. 2

From: Quantifying the degree of average contraction of Collatz orbits

Fig. 2

The theoretical quasi-stationary distribution (filled blue circles) is compared to the homologous quantity obtained from a direct numerical implementation of the deterministic Collatz map. More specifically, starting from a generic integer n, we store the mod8 representation of the numbers obtained every 3 consecutive iterations of T, before the trajectory lands on the attractive, absorbing cycle. The frequency of visits for each of the allowed classes is then reported in the figure. Red crosses are obtained running the Collatz map for all integers in between 1 and \(n_{max}=10^5\). Green squares refer to \(n_{max}=10^7\). The blue horizontal lines are drawn as guideline for the reader, the solid line corresponds to 1 / 6 and the dashed one to 1 / 12. When increasing \(n_{max}\), all entries of the quasi-stationary distribution shifts consistently, although imperceptibly, towards the theoretically predicted values. Observe that, for \(n_{max} =10^7\), the largest value attained by the system is \(M \simeq 6\times 10^{13}\) (colour figure online)

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