I was thinking about (was it Daniel?)'s idea about using the complex fixed points to compute continuous iteration of bases greater than eta, which in general would give complex solutions. For multiple fixed points (not including conjugates), the question came up as to whether the solutions would be the same. And in general, we would expect real solutions anyway, so why should we expect these complex solutions to also be correct in some way?

On the other hand, we have these bases less than 1, where continuous iteration only seems to make sense with complex outputs at non-integers.

Finally, thinking back to Henryk's analogy with exponentiation of negative bases, I made a realization. Consider exponentiation as fractional multiplication. Now consider that there are multiple complex numbers of the form that are equal to b. In the normal course of our mathematical day, we would treat these numbers as the same. But what if we momentarily considered them as different?

Then the fractional iterate of multiplication for and the fractional iterate of would be different. The first would give real values, and indeed would be what we think of as exponentiation in base b. The second would be a complex spiral. There would be multiple such spirals, with pairs spiraling in opposite directions to represent the conjugates. All would pass through the same point for a particular integer input.

For each "positive" b, there is an infinite family of values which are equal to b. Only one of them has 0 for the imaginary part (at k=0), but we get spoiled by the fact that there is a "primary" branch of exponentiation that gives us real values for real exponents. For "negative" b, there is an infinite family of values , none of which has 0 for the imaginary part. Being spoiled by the our experience with bases b = for k=0, we naively ask ourselves questions like "Why isn't there a real-valued function for negative b?" Such as silly question! (Yes, I'm being somewhat sarcastic.)

We should expect multiple exponentiation functions, as separate branches of some master relation (is "relation" the right word?). For positive b, only one of these branches returns real values for real exponents. For negative b, the best we can hope for is to pick the branch or its conjugate. Note that in these cases, we get alternating positive and negative outputs at the integers.

For complex b, the best we can hope for is to pick the branch with the smallest (absolute value) imaginary part in its logarithm. But there would always be those other branches, equally valid in some sense.

Getting back to tetration, we would expect a real-valued function for b>1, but there would likely be additional complex "branches", some of which may coincide with continuous iteration from one of the complex fixed points. For bases less than 1, we would only expect complex outputs, but there would be multiple branches (in addition to conjugates). Note that we get alternating upper and lower outputs, much as we got alternating positive and negative values for exponentiation of negative bases.

The analogy seems pretty good, but now we must dig deeper to understand the nature of it.

Edit: Heh, forgot to close all my tex tags.

edit 2: updated to include i's, and escaped the ln functions for good measure. See below evidence of my omission.

On the other hand, we have these bases less than 1, where continuous iteration only seems to make sense with complex outputs at non-integers.

Finally, thinking back to Henryk's analogy with exponentiation of negative bases, I made a realization. Consider exponentiation as fractional multiplication. Now consider that there are multiple complex numbers of the form that are equal to b. In the normal course of our mathematical day, we would treat these numbers as the same. But what if we momentarily considered them as different?

Then the fractional iterate of multiplication for and the fractional iterate of would be different. The first would give real values, and indeed would be what we think of as exponentiation in base b. The second would be a complex spiral. There would be multiple such spirals, with pairs spiraling in opposite directions to represent the conjugates. All would pass through the same point for a particular integer input.

For each "positive" b, there is an infinite family of values which are equal to b. Only one of them has 0 for the imaginary part (at k=0), but we get spoiled by the fact that there is a "primary" branch of exponentiation that gives us real values for real exponents. For "negative" b, there is an infinite family of values , none of which has 0 for the imaginary part. Being spoiled by the our experience with bases b = for k=0, we naively ask ourselves questions like "Why isn't there a real-valued function for negative b?" Such as silly question! (Yes, I'm being somewhat sarcastic.)

We should expect multiple exponentiation functions, as separate branches of some master relation (is "relation" the right word?). For positive b, only one of these branches returns real values for real exponents. For negative b, the best we can hope for is to pick the branch or its conjugate. Note that in these cases, we get alternating positive and negative outputs at the integers.

For complex b, the best we can hope for is to pick the branch with the smallest (absolute value) imaginary part in its logarithm. But there would always be those other branches, equally valid in some sense.

Getting back to tetration, we would expect a real-valued function for b>1, but there would likely be additional complex "branches", some of which may coincide with continuous iteration from one of the complex fixed points. For bases less than 1, we would only expect complex outputs, but there would be multiple branches (in addition to conjugates). Note that we get alternating upper and lower outputs, much as we got alternating positive and negative values for exponentiation of negative bases.

The analogy seems pretty good, but now we must dig deeper to understand the nature of it.

Edit: Heh, forgot to close all my tex tags.

edit 2: updated to include i's, and escaped the ln functions for good measure. See below evidence of my omission.

~ Jay Daniel Fox