This map is very suspect. China has a reputation for forging everything from art to missing link fossiles. This is just an attempt to rewrite our history in a dangerous way. It is interesting that the explorer was muslim. Is this an attempt to motivate the US Muslims to fight for the return of their holy land in the Americas? I read somewhere that this version is being taught heavily in the mosques in Mexico. I will try and find a link for it.
The official Chinese Imperial record from the time (and what is taught in Chinese history books) stated Zheng He made 7 overseas expeditions to the South Seas and all the way to East Africa (basically south to modern day Philipeans, Indonesia, Malaysia, over to India, Persia, Arabia, and East Africa)- these are well documented trips supported by ample formal historcal records and archealogical evidence.
Pretty much everything else is conjucture of what other places Zheng's fleet *might* have reached.
I don't see why the fact that Zheng He was, historically, in fact a Muslim should twist anyone's panty into a wad.