@Gottfried: By which method did you compute your values?
Just triggered by the question about b[4]I, I computed it by \( b[4]t=\exp_b^{\circ t}(1) \) with \( \exp_b^{\circ t} \) being the regular iteration at the lower fixed point of \( b^x \) (via the formula given here) for various bases \( 1<b<\eta \):
This picture has also the disadvantage that you dont see which base \( b \) is associated to which point on the curve.
So I add a list of values:
\( b \) , \( b[4]I={exp_b}^{\circ I}(1) \)
There is also this interesting base \( b>1 \) for which \( b[4]I \) is real.
Just triggered by the question about b[4]I, I computed it by \( b[4]t=\exp_b^{\circ t}(1) \) with \( \exp_b^{\circ t} \) being the regular iteration at the lower fixed point of \( b^x \) (via the formula given here) for various bases \( 1<b<\eta \):
This picture has also the disadvantage that you dont see which base \( b \) is associated to which point on the curve.
So I add a list of values:
\( b \) , \( b[4]I={exp_b}^{\circ I}(1) \)
Code:
1.01, 1.011233887 - 0.01003728120 I
1.02, 1.035202770 - 0.01405751590 I
1.03, 1.059991495 - 0.01057562160 I
1.04, 1.083268505 - 0.002332372600 I
1.05, 1.104485443 + 0.009021821700 I
1.06, 1.123581670 + 0.02245648730 I
1.07, 1.140655009 + 0.03729317620 I
1.08, 1.155854058 + 0.05306607940 I
1.09, 1.169338195 + 0.06944512180 I
1.10, 1.181261981 + 0.08619082190 I
1.11, 1.191769271 + 0.1031263661 I
1.12, 1.200991438 + 0.1201195506 I
1.13, 1.209047362 + 0.1370706818 I
1.14, 1.216044168 + 0.1539042223 I
1.15, 1.222078233 + 0.1705628566 I
1.16, 1.227236250 + 0.1870032102 I
1.17, 1.231596256 + 0.2031926760 I
1.18, 1.235228576 + 0.2191070431 I
1.19, 1.238196668 + 0.2347287000 I
1.20, 1.240557891 + 0.2500452354 I
1.21, 1.242364161 + 0.2650483724 I
1.22, 1.243662552 + 0.2797331122 I
1.23, 1.244495805 + 0.2940970757 I
1.24, 1.244902798 + 0.3081399627 I
1.25, 1.244918930 + 0.3218631346 I
1.26, 1.244576497 + 0.3352692637 I
1.27, 1.243904987 + 0.3483620634 I
1.28, 1.242931367 + 0.3611460599 I
1.29, 1.241680334 + 0.3736264086 I
1.30, 1.240174520 + 0.3858087482 I
1.31, 1.238434699 + 0.3976990769 I
1.32, 1.236479952 + 0.4093036533 I
1.33, 1.234327829 + 0.4206289147 I
1.34, 1.231994485 + 0.4316814110 I
1.35, 1.229494806 + 0.4424677512 I
1.36, 1.226842509 + 0.4529945543 I
1.37, 1.224050270 + 0.4632684218 I
1.38, 1.221129787 + 0.4732959017 I
1.39, 1.218091867 + 0.4830834635 I
1.40, 1.214946518 + 0.4926374870 I
1.41, 1.211702993 + 0.5019642400 I
1.42, 1.208369868 + 0.5110698727 I
1.43, 1.204955087 + 0.5199604031 I
1.44, 1.201466009 + 0.5286417061 IThere is also this interesting base \( b>1 \) for which \( b[4]I \) is real.
