\begin{tikzpicture}%[every node/.style={scale=0.7}]

% configurable parameters
\def\eps{0.1};
\coordinate (ub) at (+4.0,0);
\node at (ub) [right] {$+\infty$};
\coordinate (lb) at (-3.0,0);
\node at (lb) [left] {$-\infty$};
\coordinate[label ={[xshift=3*\eps cm, yshift=\eps cm]$t_0$}] (t0) at ($(-1.0,0)$) {};

% partition
\def\het{0.5};
% \draw[dotted] (-1.0, -\het)--(-1.0, \het);
\path[fill=black, fill opacity=0.1] ($(lb) + (0.0, -\het)$) rectangle ($(-1.0, 0.0) + (0.0, +\het)$);

% axes
\draw [->,help lines] (-2.5, 0)--(3.5,0) node[below]{$t$};

% path
\draw[
    densely dotted,
    thick,
    decoration={markings, mark=at position 0.625 with {\arrow{>}}},
    postaction={decorate}
    ] ($(lb) + (0.0, \eps)$) to ($(ub) + (0.0, \eps)$);

\draw[
    densely dotted,
    thick,
    ] ($(ub) + (0.0, \eps)$) to[out=-30,in=+30] ($(ub) + (0.0, -\eps)$);

\draw[
    densely dotted,
    thick,
    decoration={markings, mark=at position 0.375 with {\arrow{<}}},
    postaction={decorate}
    ] ($(lb) + (0.0, -\eps)$) to ($(ub) + (0.0, -\eps)$);

% dots
\node at ($(lb)+(0,\eps)$)[circle,fill,inner sep=0.8pt]{};
\node at ($(lb)-(0,\eps)$)[circle,fill,inner sep=0.8pt]{};

\node at ($((-1.0, 0.0)+(0,\eps)$)[circle,fill,inner sep=0.8pt]{};
\node at ($((-1.0, 0.0)-(0,\eps)$)[circle,fill,inner sep=0.8pt]{};

\end{tikzpicture}