Ancient cultures certainly influenced each other. One developed a skill or tools and, through trade with others, the innovation was passed on.
The Anasazi had commerce with other major cultures in the Four Corners Area, like the
Hohokam and Mogollon. They all influenced each other as well as the development of other cultures, like the Sinagua and Salado.
After the Anasazi left their great houses and cliff dwellings in the 12th and 13th centuries, their culture emigrated with them. Thus, their influences can be seen even today in the Hopi, Zuni and Pueblo cultures.