Construction of Mandelbrot Set

Question

I am doing Master's project in Complex Dynamics. Here I want to talk particularly about the Mandelbrot set. I have studied its formation and dynamics as parameter $c$ changes (which is very hand waving in it) from book called "Iteration of rational function" by Alan F. Beardon and some few properties of it given in this book in chapter $9$ section $9.10$.

I need references for the following things

$(1)$ I want to learn more rigorously about its construction.
$(2)$ Also want to understand dynamics of its boundary.
$(3)$ Its connection with complex logistic map.
$(4)$ Fractal nature of Mandelbrot set.

Answer 1

Perhaps, you may read "The Mandelbrot set, theme and variations" by Tan Lei. You will be interested in reading it. It discusses the universality of the Mandelbrot set.

Answer 2

Another reference is Carleson-Gamelin's book "Complex dynamics". Chapter VIII is devoted to the Mandelbrot set.

Answer 3

There are many books introducing to complex one-dimensional dynamics. My favorite one is Milnor, Dynamics in one complex variable. It is available on the arXiv.