### 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 m_{0},...,m_{d-1} and
d integers x_{0},...,x_{d-1}.
Then T(x)=int(m_{i}x ⁄ d)+x_{i}, 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 m_{0}=1, m_{1}=3, x_{0}=0, x_{1}=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(m_{i}, d) = 1 for 0 ≤ i ≤ d-1, or

(b) gcd(m_{i}, d^{2}) = gcd(m_{i}, d) for 0 ≤ i ≤ d-1 and d divides m.

Here Q_{i}_{j}(m)=p_{i}_{j}(m)/d,

where p_{i}_{j}(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.

*Last modified 21st July 2011*

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