### David Barina's mod 8 conjecture

The iterates x, t(x), t(t(x)),... of the mapping

t(x) | = | 5x+1 | if x ≡ -1 (mod 8) |

t(x) | = | 5x-1 | if x ≡ 1 (mod 8) |

t(x) | = | ⌊x/2⌋ | otherwise |

are conjectured to eventually reach 1 if x > 0, or one of -1 or -31 if x < 0.
This phenomenon was communicated to Keith Matthews by David Barina on July 13, 2018.

The mapping can be regarded as an 8-branched example of type (b).

Here Q(8) has the zero class as a transient class and the submatrix formed
by deleting rows 1 and column 1 has stationary vector
(1/23)×(3,4,2,5,3,4,2).

The corresponding weighted product is 5^{3/23}(1/2)^{4/23}(1/2)^{2/23}(1/2)^{5/23}(1/2)^{3/23}(1/2)^{4/23}5^{2/23} < 1

⇔ (1/2)^{(4+2+5+3+4)/23}5^{5/23} < 1

⇔ 5^{5} < 2^{18} ⇔ 3125 < 262144,

thereby predicting everywhere eventual cycling. The cycles found are

- 0
- -1, -4, -2
- 1, 4, 2
- -31, -156, -78, -39, -196, -98, -49, -244, -122, -61.

*Last modified 27th July 2018 *

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