Does anyone know of a period for which there are more bulbs than cardioids?

2 (one bulb, no cardioids) and 3 (two bulbs, one cardioid). 4 is a tie (three bulbs, three cardioids), 1 has a cardioid and no bulbs, and at 5 there are four bulbs and 11 cardioids. The cardioids explode from 6 on upward.

On one side, there are countable infinitely many bulbs and as well cardioids. So in a sense, they are "equal in numbers". On the other hand, every minibrot consists of one cardioid and infinitely many attached bulbs, so in that sense, there are vastly more bulbs.

Bulbs occur in arithmetic progressions. Start with a cardioid or a bulb and its bond point cusp. About midway between the cusp and itself will be a single bulb the sum of the relevant periods (so, 2x the period of the parent in this case). That bulb's bond point becomes a second cusp. Between this cusp and the parent bond cusp, going in either direction, there will be a single bulb the sum of the relevant periods (so this time, 3x the period of the parent). That gives two more cusps, and four more points midway between pairs of cusps, and so forth.

So if you have a bulb, its period is a sum of adjacent larger bulbs or parent-component cusps that all have smaller periods. Any larger bulbs have the same property. So the period of a bulb is just the sum of the parent component period with itself some number of times. The way that sum is parenthesized describes the specific bulb, relative to its parent. So, for a bulb of period 5 directly on the main cardioid that sum is 1 + 1 + 1 + 1 + 1. If we parenthesize it as 1 + (1 + 1 + 1 + 1) then it's between a period 1 bond point and a period 4 one. So, between elephant valley and a period 4 bulb. If we parenthesize it as (1 + 1) + (1 + 1 + 1) it's between bulbs of periods 2 and 3, so the starfish bulbs above and below the seahorse valleys. Those are the only two places to split the sum into two smaller parts, plus their symmetric counterparts, so there's just the four bulbs.

For a generic period

*p*, there will be a number of bulbs about equal to

*p* itself on the main cardioid. (Actually, equal to the number of invertible numbers mod

*p*, in practice; so, never more than

*p* - 1 and usually not much less than that.) For bulbs of period

*p* elsewhere, for each divisor

*k* of

*p*, there will be about

*p/k* bulbs of period

*p* directly on each component of period

*k*, the latter numbering a bit fewer than 2

^{k-1}. Out of the 2

^{p-1} components of period

*p*, then, the number of bulbs is bounded above by \( \sum_{k:k|p}\frac{p}{k}2^{k-1} \). Since divisors are fairly sparse, and get sparser the bigger a number is (scaling as the logarithm of that number), this will tend to be well below the sum over

*every k* <

*p* of 2

^{k-1}, which is 2

^{p-1} - 1, which is the number of components of period dividing

*p* in total, less one. The "missing" components all have to be cardioids. In particular,

*p* - 1 doesn't divide

*p* when

*p* > 2 and contributes half of the latter sum, and

*p* - 2 doesn't divide

*p* when

*p* > 4 and contributes another quarter, so for

*p* > 4 at least half and typically over 3/4 of the components are going to be cardioids.

So, there is a sense (counting components of a period) where cardioids vastly outnumber bulbs.

This all goes to show how counterintuitive infinite sets can be. Whether there's more of one infinite subset or another usually depends on the order they get counted in (always, when the subsets have the same infinite cardinal, and in particular when the subsets and parent are all countable and infinite).

What would happen to the ratio of bulbs up to a certain period divided by the number of cardioids? Will that value have a limit, grow unboundedly or simply jumping around?

Long run? Asymptotic to zero. The largest proper divisor of a number is no more than half of it, so the bulbs are bounded above (usually very loosely) by

*p*(1 + 2

^{p/2 - 1}), which is order of sqrt(number of components of period

*p*) for large enough

*p*, which makes the bulb to cardioid ratio stay below 1/sqrt(number of components of period

*p*) past a fairly small

*p*, empirically determinable to be about 5.