The Aztlan Myth
The Aztlan Myth is the belief that the Mexican government or the native population of Mexico is entitled to ownership of the territory that is now the western and southwestern USA (California, Arizona, New Mexico, Texas).
Archaeologists long ago determined that excavated ancient artifacts, structures and tools or languages identified as indigenous to those areas far more-closely matched ones derived from tribes which migrated down from the NORTH (across the frozen Bering Strait 10,000 years ago or so when it connected Alaska to Siberia). The artifacts/languages which derived from cultures of South/Central America were completely different from those found in the southwestern U.S. territories.
The reason people from South/Central America never settled in large numbers in the southwestern U.S. territories is obvious - the southern borders are all wide, desolate deserts. It was impossible for any culture from South America or Mexico to transport large populations, warriors or major provisions across hundreds of miles of that virtually-waterless, oppressive desert. Any large group attempting passage through the deserts with the technology available in those times would have died long before they got further north to regions which could have supported cities. So the scant tribes which had permanent settlements there were the same ones who inhabited the rest of the USA (all migrated down from the north).
The only claim of right to the territory would then be international law or accepted moral conventions of civilization.
Since modern international laws recognizing national sovereignty did not exist in the 1820s when the Mexican-American War took place, the separation of those territories represented an exercise in an ancient law which derives more from the Natural Law of the Enlightenment than any other - those who are able to defend their territorial borders have the right to dominion - and those who have the right to dominion are only those who can defend their borders.
At the time of the Mexican-American War, in California there were at least as many non-Mexicans living in California as Mexicans. The total Mexican population consisted of a few minuscule forts and the families that worked the ranchos, likely not more than 2 thousand at most. That population was matched in equal or surpassing numbers by U.S. fur trappers, traders, settlements and other foreign nationals from Russia, England, Germany, Spain, etc.
The soldiers posted at the fort maintained by Mexico near what is now San Francisco were easily defeated by a small force of U.S. Cavalry which had traveled on horseback or walked most of the way across the country from the Midwestern U.S. territories. The roughly 60 Mexican soldiers posted surrendered without a fight.
There were so few Mexican soldiers in California when the news arrived that the U.S. Cavalry had captured the fort, the Mexican government had to send soldiers all the way from Mexico to defend their interests - there were no large Mexican garrisons in California at all circa 1845.
This is significant because the U.S. government was fully aware that other nations (Britain, Germany, Russia, France) had immediate and rapidly-growing interests in possibly invading / fortifying California's territory and using the natural harbors as bases for significant invasion forces.
The U.S. offer to buy California and other territories from Mexico was rejected by their government, so the U.S. government decided that it had little choice but to move into those territories and set up forts / settlements in order to counter inevitable foreign incursion into the region.
So there never were large developments, towns or even settlements by Mexico in either California or any of the other states when the Mexican-American War started. Many of the Mexican nationals who lived in the disputed territories ended up fighting for the U.S. because they had little loyalty to the Mexican government which only had contact with them when they sent soldiers to take the horses that had been broken, trained and bred at the rancheros back to Mexico along with taxes and other things of value. The Mexican government / military provided little or no protection to the Mexican nationals living in California - they were mostly left to defend themselves from hostile native tribes in the area.
Once the war movement started, many of these led by a Mexican military man, called themselves the Californios and fought along side the Cavalry for independence from Mexico.
The last remaining claim to rightful "ownership" by Mexico would be the antiquated claim that Spanish Conquistadors traditionally laid to any land which harbored their ships, "We claim in the name of God and the King of Spain all lands north, south, east and west as far as a conquistador can lead a horse". So in essence, they laid claim to the entire North American Continent simply because they landed ships on the shores of Baja California and the Gulf of Mexico.
So you can see that there is absolutely no legitimate ancestral, moral, legal or practical claim of ownership that the Mexican government or the peoples of Mexico can make on the western and southwestern United States. The USA did not "steal" so much as "liberate" the states from Mexico and in an act of almost unprecedented graciousness after winning the conflict, voluntarily paid the Mexican government a hefty sum of money as compensation for taking the land from them.
Since that time US cities have provided millions of jobs for Mexican nationals through three centuries while their own nation struggled with massive institutional corruption, unemployment and pollution problems.
The USA has diligently helped to defend Mexico militarily through two world wars and on into the present day without receiving compensation of any kind. Our soldiers died by the millions defending not only ourselves, but indirectly Mexico time and time again without compensation from or participation by them directly.
U. S. companies, government and private capitalists have invested countless trillions of dollars in the infrastructure and economic development of the western/southwestern states since the 1820s while the Mexican people and government could barely scrape together the resources to develop and administrate their own existing territories, much less invest in developing the U.S.A.'s western and southwestern states.
So, as you can see, the claim that the U.S.A. "stole" what some refer to as Aztlan is absolute, utter, total and complete anti-USA, anti-white, Hispano-centric bull.
The indigenous Mexican peoples (Aztecs, Toltecs, Yaquii, etc.) never dwelt in large numbers in the western/southwestern U.S. territories, neither in the 1800s nor in ancient times.
The Mexican government / indigenous peoples have no morally or legally valid claim to these territories. The only laughably silly basis for such a claim is a wholly-symbolic, legally-unenforceable, inherited boiler-plate declaration made by Spanish explorers/Conquistadors in the 15th century. Native Mexicans never established major cities or infrastructure in those territories in the modern era and therefore have no rightful claim to dominion.