My question concerns a quite elementary problem in set-theoretic topology: Assume that $(X,d)$ is a compact metric space. Consider the infinite product $X^{\mathbb{Z}}$ of all (two-sided) sequences in $X$. Besides the product topology, there are at least two other topologies on $X^{\mathbb{Z}}$ induced by metrics which are very natural:

(1) The topology induced by the sup-metric: $D_1(x,y) = \sup_{n\in\mathbb{Z}}d(x_n,y_n)$.

(2) The topology induced by the metric $D_2(x,y) = \sup_{n\in\mathbb{Z}}D_0( \theta^n x,\theta^n y)$, where $D_0$ is any metric on $X^{\mathbb{Z}}$ that induces the product topology and $\theta:X^{\mathbb{Z}} \rightarrow X^{\mathbb{Z}}$ is the left shift operator that sends a sequence $(x_n)_{n\in\mathbb{Z}}$ to $(x_{n+1})_{n\in\mathbb{Z}}$ ($\theta^n$ is the $n$-th iterate of $\theta$). The metric $D_0$ may be given by $D_0(x,y) = \sum_{n\in\mathbb{Z}}\frac{1}{2^{|n|}}d(x_n,y_n)$.

My question is: Are there any known characterizations of connectedness of $(X^{\mathbb{Z}},D_i)$, $i=1,2$, in terms of the properties of $(X,d)$?

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