The phrase "dominating influence" means having control or power over someone or something, being the most powerful or influential force in a particular situation. Full definition
The conditions also bolster the views of ice and climate specialists who have stressed that the many factors shaping Arctic conditions year by year, from winds and atmospheric pressure to highly variable ocean currents and soot, still dominate the influence of heat trapped by building greenhouse gases. (dotearth.blogs.nytimes.com)
It'd be amazing to add yet another first party studio to the Nintendo umbrella of w̶o̶r̶l̶d̶ console dominating influence. (nintendotoday.com)
In particular, the authors find fault with IPCC's conclusions relating to human activities being the primary cause of recent global warming, claiming, contrary to significant evidence that they tend to ignore, that the comparatively small influences of natural changes in solar radiation are dominating the influences of the much larger effects of changes in the atmospheric greenhouse gas concentrations on the global energy balance. (climatesciencewatch.org)