Consider a hyperbolic 3-manifold, called a mapping torus, that fibers over a circle with fiber a closed orientable surface of genus at least 2. Cannon and Thurston showed that the inclusion map from the surface into the 3-manifold extends to a continuous, surjective map between the visual boundaries of the respective universal covers. This gives a surjective map from a circle to a 2-sphere. Mj showed that a Cannon-Thurston map also exists for a hyperbolic group and its normal hyperbolic subgroups. In this talk, we will explore what happens when we consider the mapping torus of a surface with boundary, which is not hyperbolic but CAT(0) with isolated flats under some conditions.