There is a related iterated function system with two functions,

$f_0(x) = 1+zx$

$f_1(x) = -1+zx$

$X_z$ is the unique nonempty compact fixed set of this iterated function system. It is sometimes called a generalized dragon set with parameter $z$, and particular values of $z$ can produce some well-known fractals called dragons.

A relevant result on iterated function systems is that the fixed set $X$ is connected iff it is arcwise connected iff the family of subsets $\{f_i(X)\}$ is connected, which in this case means $f_0(X_z) \cap f_1(X_z)$ is nonempty. ([This paper][1] refers to  Kigami, *Analysis on Fractals* chapter 1 for the result.) So, the set is connected iff we can write 

$$\begin{eqnarray} 1 + \sum_{i=1}^{\infty} a_i z^i &=& -1 + \sum_{i=1}^\infty a_i' z^i \newline 1 + \sum_{i=1}^{\infty} \frac{a_i-a_i'}{2} z^i &=& 0 \newline 1 + \sum_{i=1}^{\infty} b_i z^i &=& 0 \end{eqnarray}$$

where $b_i = (a_i-a_i')/2 \in \{0,1,-1\}$.

In particular, $X_z$ is connected when $z$ is real with $1/2 \le |z| \lt 1$ and when $z$ is a root of a polynomial with coefficients in $\{-1,0,1\}$. The intersection of the closure of those roots with the interior of the disk is the entire set where $X_z$ is connected.

[![enter image description here][2]][2]

This image shows the nonzero roots of polynomials of degrees up to $9$ with coefficients in $\{-1,0,1\}$ with the unit circle.

The closures of roots of polynomials with restricted coefficients have been studied, and they are [quite interesting][3]. In some areas, there seems to be a Julia-Mandelbrot correspondence, where the set of roots of small degree near a point resembles the fixed set of the iterated function at that point. However, the pictures for a fixed small degree may be misleading. [It seems to be known][4] that the entire annulus $1/\sqrt{2} \lt |z| \lt 1$ is contained in the closure of the roots of polynomials with coefficients in $\{-1,0,1\}$.


  [1]: http://www.math.uaic.ro/~annalsmath/pdf-uri%20anale/F1-2009/Alexandru-Mihail.pdf
  [2]: https://i.sstatic.net/aoprH.png
  [3]: http://math.ucr.edu/home/baez/roots/
  [4]: http://topo.math.u-psud.fr/~bousch/preprints/clh_ifs.pdf