BOMP - Ch 19- Breathing
The simplest, most human of activities, we breathe about 800 million times during an average lifespan, mostly without thinking about it. I think that this is also perhaps the most difficult of principles to get a handle on.
Certainly breath is related to other rhythms in life and in martial arts. Research has shown that elite runners and rowers learn to spontaneously entrain, or synchronize, their breathing with the rhythm of their activity. The same research shows that entrainment of breathing reduces perception of effort, but it is not shown to reduce actual energy expenditure.
Breath is intuitively important. It feels like something we should concentrate on. But it is such a slippery concept that every supposed guru that you can find will tell you to control your breath in different ways. For that reason, whenever I've had a student ask me for specifics about breath control, I've mostly told them, "Do this thing a million times and try to get to the point that it doesn't make you too out of breath to do it vigorously."
In other words, pay attention to other stuff for a long time, and your own subconscious will figure out how to best control your own breathing in acceptably good synchrony with your own biomechanics.