Dimension of the accumulation set of any hair for the exponential map

We study the dynamics of the exponential map on the complex plane. The set ${\mathrm{\Lambda }}_{\mathbf{c}}$ of all points sharing a given itinerary $\mathbf{c}$ is non-empty if and only if $\mathbf{c}$ is an exponentially bounded itinerary. For such itineraries, ${\mathrm{\Lambda }}_{\mathbf{c}}$ also contains a curve of escaping points, and hence its Hausdorff dimension is at least~$1$. We prove that for every exponentially bounded itinerary this dimension is in fact equal to~$1$. In comparison, for certain itineraries, the set ${\mathrm{\Lambda }}_{\mathbf{c}}$ exhibits highly complicated topological structures, such as indecomposable continua.