Jim, the answer may differ based on the different areas. In some northern areas, deer and elk tend to rut earlier by a few weeks than they do in southern regions. Here in Colorado, the rut will usually get hot and heavy around the 2nd or 3rd week of September.
Hiker is dead on for peak rut in Colorado. It varies by latitude, like whitetail. I'm not for sure, but I believe Montana and Canada are a few weeks earlier (at most) than Colorado and New Mexico or Arizona can be a few weeks later.
Any forum members want to comment on their region?
The record books, that are put out by Boone & Crocket and Pope & Young and where I live is Henry Kelsey they are a great way to research a state or a province for big game. This does take some time but the knowledge it gives you, can be very valuable. Not all hunters have their trophies recorded, as where I live after the Hanson Buck became the new world record typical Whitetail, we saw a flood of hunters and now that has happened, some hunters are shying away from the record books but...