They are just big kitty-cats, so you don't want to appear as smaller prey to them. In particular, running away makes them think you are prey, and will encourage an attack. Do not take your dog with you into the wilderness, if you want to reduce your chances of a cougar attack. They feel they will be protected from cougars. But that's not the case.
Dogs are an attractant in most cases.
However, you may not have to worry about taking action to prevent an attack, since mountain lions ordinarily either lie hidden, waiting for prey to approach beneath them, or approach unseen, and then attack and kill by a bite to the back of the neck that severs the spinal cord. This was the modus operandi for the attack on Barbara Schoener . To Survive A Cougar Encounter . Statistics of Attacks These are quotes or summaries from various sources that have given numbers on the total number of attacks within a given period. Note that the information is only current to the date of the reference .