بررسی رفتارهای بحرانی و وجود جاذب های عجیب در خانواده ای که یک پارامتری از دیفیومورفیسم ها روی دایره واحد

in chapter one we will describe definitions and preliminary results to provide the global context of our own results to be presented in detail in the subsequent chapters in chapter two we consider degree-one maps of the circle and we study their rotation set. our main result in this chapter says that if the map is topologically mixing then its rotation interval is nontrivial (that is, not reduced to a point) and nonper-sistent (arbitary small pertubations may modify the rotation interval). non-persistenence is a very important property because, by a result of newhouse, palise and takens [n.p.t-1983], modification of the rotation inveral always involve homoclinic bifurcations and so also formation of chaotic dynamics. in chapter three we consider a class of one-parameter fmilies of degree-one maps of the circle and study the presence of strange attractors in these families. more precisely, we show that the derivative grows exponentially fast on the critical orbits for a set of parameter values with positive lebesgue measure. evenmore, this set of parameters has full lebesgue density at the special bifurcation point. then the closure of the critical orbits is a strange attractore for theses maps.