How are the Arctic natives of America similar to those of Eurasia?
Do you mean you want a comparison of Inuit/Yupik peoples in Alaska and eastern Siberia?
I know, for one, that they all speak languages descended from a common ancestor, all of which languages are included in the Eskimo-Aleut family. This would imply a common origin. Alaskan Inuit are also distinct from American Indians in Canada, the United States, etc., and I believe this is because they came over at a later time than the ancestors of those other groups.
EDIT: maybe this will help too: http://en.wikipedia.org/wiki/Russian_Far_East#Demographics; http://en.wikipedia.org/wiki/Alaskan_native
I know Wikipedia isn't the most reliable source, but there are usually good links to reputable sites in the Bibliography section at the bottom.
Have some Arctic cultures (including languages) adapted differently according to their similar environments? For instance, how do far-northern Scandinavians compare to other Arctic peoples?
Philosoraptor, your own comments are much more informative than Wikipedia.
Laplanders are interesting and some look similar to Eskimos in their stature and faces, although the clothing is different.
I think Arctic cultures are very different. The eastern Siberian and Alaskan peoples are quite similar, for the reasons I mentioned above, but going around the globe you get as much difference as I imagine you'd find in civilizations around any given latitude. The ancient Egyptians and Aztecs were at roughly the same latitude, and there are no real similarities in their cultures; I don't see any reason why Scandinavian peoples should have similarities with those in the far east of Russia.
The comparison with the Scandinavians may be faulty though, because the climate where most northern Europeans live is very different from the arctic. A better comparison might be the Sami people of northern Norway (http://en.wikipedia.org/wiki/Sami_people), but I don't know enough about them to really say. One thing to take into account would be the differences in natural resources, flora, and fauna between the areas. For instance, caribou inhabit a large part of the North American polar wilderness, but I don't believe they're found in many of the other polar areas. Their presence would certainly influence the lifestyles of semi-nomadic peoples in ways different from those area which don't have them.
No doubt there are some similarities between polar people--as there are, to some degree, between all people of similar climate. But I don't think these would be noticeable enough to make many blanket statements about them, without being overly reductionist.
EDIT: Oops, I see Evo already mentioned the Sami (aka Laplanders). My bad!
EDIT 2: On second thought, the caribou example may not be a good one. Looks like caribou are more widespread in polar regions than I thought. However, I think my point stands... just substitute in some species unique to a given area (this goes for plant life, too) for 'caribou'
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