# Hausdorff Dimension of Imploded Julia Sets

To within 0.1:

• Cauliflower: what is $$\liminf_{ε→0}[\text{dimJ}(0.25+ε,0)]$$? Or let $$λ$$ be the external ray with argument 0, compute $$\liminf_{λ↓}[\text{dimJ}]$$?
• San Marco: let λ have argument ±⅓, compute $$\liminf_{λ↓}[\text{dimJ}]$$?
• Fat Rabbit: let λ have argument ±⅐ or ±²⁄₇, compute $$\liminf_{λ↓}[\text{dimJ}]$$?
• In these cases, how does the $$\liminf$$ dimension increase if we travel perpendicularly to each ray and approach the boundaries of the valleys?

Edit: Context

Both Shishikura 1991 and Buff-Chéritat 2005 looked at parabolic implosion to find Julia Sets with dimension arbitrarily close to 2. I just wondered what the dimension can be due to a single parabolic implosion, regarding the first three parabolic points in the Mandelbrot Set? e.g. McMullen 1997 calculated the dimension $$\text{dimJ}(0.26,0)\approx 1.3355$$ in the Appendix.

Using a computer program, Claude Heiland-Allen calculated "Seahorse Dust" (Slide 159 in https://mathr.co.uk/mandelbrot/julia-dim.pdf) with box-counting dimension of 1.609, which is most likely an overestimate. Nonetheless, the dust is very loose; so the imploded Seahorse may have a dimension of 1.7? And the imploded Fat Rabbit 1.8?

GIF: Cauliflower, San Marco, and Fat Rabbit imploded Julia Sets of increasing complexity.

Since nobody answered the question for a whole month, I will answer:

So far, I wasn't able to find any literature on the numeric Hausdorff dimension of a Julia Set on the imploded cases; see Hausdorff Dimension of Arbitrary Julia Set. Therefore, I will use Claude 2015 in the answer. Claude Heiland-Allen numerically approximated the box-counting dimension of Julia Sets over the Mandelbrot Set. Below is his image:

Using estimation based on the colors of the image:

• Cauliflower: 1.36 in the middle to 1.62 on the boundary of valley.
• San Marco: 1.60 in the middle to 1.66 on the boundary of valley.
• Fat Rabbit: 1.69 in the middle to 1.72 on the boundary of valley.

For additional comparison, the Julia Set with Misiurewicz point of period $$n$$ in the Elephant Valley has dimension around 1.83 (below left), and a parabolic implosion within an elephant valley bulb has dimension around 1.87 in the middle of the valley to 1.94 in the main densest spirals cardioid-side (below right). So the dimension approaches 2 rather quickly (inspired by Shishikura 1991).

Context

The 1st source constructs a Julia Set with dimension arbitrarily close to 2 using continued fractions on the cardioid. The 2nd source shows the Feigenbaum Julia Set has dimension less than 2. The 3rd source computes dimensions of Julia Sets on the real line to the left of the Feigenbaum point. The 4th source shows parabolic implosion increases the dimension drastically. The 5th source shows the existence of a Julia Set of dimension 2.

Remarks:

• Siegel Disk Julia Set has dimension around 1.66. Main Misiurewicz points in the vicinity have dimension close to 2 (>1.99).
• Feigenbaum Julia Set has dimension around 1.57. D.-Gorbovickis-Tucker 2021 showed a lower bound of ≥1.49781.
• Minijulias of high period minibrots are complex, but don't add to the dimension. e.g. The dimension is the maximum of all relative seeds (of previous minibrots passed) and the original seed. According to and Mandel Help File 6 page 10 (from Wolf Jung 2007):

In the limit n → ∞, the decorations get dense in the plane, while their area goes to 0. They get thinner, and not more hairy as in the Feigenbaum case (cf. page 10 of Chapter 5).

• Avila and Lyubich 2015 did numeric computation on general Feigenbaum maps, and posited that the map to the period-2207 primitive minibrot above the 15th bulb in the golden-mean Siegel sequence, could have Feigenbaum map with dimension 2 (below, circled in blue).

• McMullen 1995 showed that a general Siegel disk has asymptotic similarity around the origin. Also Buff and Chéritat 2008 used this method to find a Julia Set with dimension close to 2. For the Golden Mean, two dilations is around $$1.8176<\phi^2$$, where as for successive parabolic implosions, one dilation is around $$e^{1.3482}\approx 3.85<\infty$$, both shown below. This shows Julia Sets are compressive around Siegel and parabolic points, e.g. when zooming persistently along the cardioid, as opposed to the dilation at the Feigenbaum point ($$2.5029>\sqrt{4.6692}$$).