TRAjectories

Definition

Let Q be a trajectory beginning with x₀ :

Q = {x_k}, k Î [0,s_¥(x₁)], x_k+1= T(x_k)

We note Q_i the partial trajectories :

Q_i = {x_k}, k Î [0,i]

And t_i the associated thread :

Properties

P1 :

A trajectory takes values on positive or negative integers only depending of the sign of x₀

P2 :

Click here for a demonstration

Repartition in modulo 2 classes

The repartition of the x_k among the congruence classes modulo 2 is O(x₀) odd and E(x₀) even :

p_0,2 = O(x₀)/s_¥(x₀) = c

p_1,2 = E(x₀)/s_¥(x₀) = 1-c

Repartition in modulo 3 classes

To evaluate the repartition of the x_k among the congruence classes modulo 3, let us look at the forward steps for congruence classes modulo 6 :

The probabilities are such as :

With :

Solving the system we have :

Of course, this repartition is for infinite (or very long) trajectories, for real (or smaller) trajectories we musttake into account that they can begin with one or several numbers in [0]₃

The first number as a probability of 1/3 to be in [0]₃, the next of 1/2 if the first is in [0]₃ and so on.

The average quantity of numbers in [0]₃ in a trajectory is then :

The probability to be in [0]₃ is then n_0,3/s_¥ and as the height, the Total Stopping Time and the completeness verify,

The probability to be in [0]₃ is :

The repartition for a given height and a given completeness is then :

For example, the trajectories of numbers between 10000 and 20000 gives the real repartitions and the model as follows :

Figure 1 : Equivalence Classes modulo 3
among trajectories of given completeness c