Over‐Rotation Pairs ( pr  + 1, ( pr  + 1) m  +  r ) for Periodic Orbits of Unimodal Expanding Self‐Maps on the Interval

Let m ≥ 3, r ≥ 2 and p be positive integers. Write P ( x , m ) = ( x m − 2 x m −1 + 1)/( x m −1 ( x − 1)) and F m , p , r ( x ) = x p m +2 ( x p m − x m )( x ( r − 1)( p m + 1) − 1) + x ( x p m − 1)( x p m +1 − 1) P ( x , m ) + x p m +2 ( x m − 1)( x r ( p m + 1) − 1) P ( x , m + 1). In this paper, we prove that if f is an unimodal expanding self‐map on the interval with an expanding constant λ satisfying F m , p , r ( λ ) ≥ 0, then f has a periodic orbit with the over‐rotation pair (pr + 1, (pr + 1) m + r ). In addition, this paper also points out that when F m , p , r ( λ ) 1 but no periodic orbit with an over‐rotation pair (pr + 1, (pr + 1) m + r ).