Generalized 3x+1 functions and Markov matrices

We study a function T: ℤ→ℤ with d (> 1) branches, defined in terms of d non-zero integers m0,...,md-1 and d integers x0,...,xd-1.

Then T(x)=int(mix ⁄ d)+xi, if x≡ i (mod d).

Example. The 3x+1 mapping T(x)=x ⁄ 2 if x is even, (3x+1) ⁄ 2 if x is odd, corresponds to m0=1, m1=3, x0=0, x1=1.

Starting in 1982, Tony Watts and I investigated the frequencies of occupation of congruence classes (mod m)
of trajectories arising from generalized 3x+1 functions. (See earlier 3x+1 interests)
Experimentally we observed in cases (a) and (b) below, that divergent trajectories were attracted to ergodic sets (mod m)
of a Markov matrix Q(m) and with predictable frequencies. (See properties of Q(m).)

(a) gcd(mi, d) = 1 for 0 ≤ i ≤ d-1, or
(b) gcd(mi, d2) = gcd(mi, d) for 0 ≤ i ≤ d-1 and d divides m.

Here Qij(m)=pij(m)/d,
where pij(m) is the number of congruence classes (mod md)
which comprise B(i,m) ∩ T-1(B(j,m)),
where B(i,m) denotes the congruence class of integers x ≡ i (mod m).

The present program constructs Q(m) and finds the ergodic sets and transient classes, using the Fox_Landi algorithm with labels 0,1,...,m-1.

In the case when neither conditions (a) nor (b) hold, it is still possible to get conjectural information using Markov matrices and this was pointed out by George Leigh in 1983.

Enter d (1 < d ≤ 20):
Enter m (1 < m ≤ 200):
Enter the d non-zero integers mi (separated by spaces):
Enter the d integers xi (separated by spaces):

Last modified 21st July 2011
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