Dynamical properties of maps derived from maps with strong negative Schwarzian derivative (Q1078683)

Let \(f: R\to [0,1]\), \(g: R\to [0,+\infty)\) have continuous third derivatives in [0,1] and in (0,1), respectively, \(f(x)=0\) for \(x\not\in (0,1)\), \(g(x)=0\) for \(x\not\in [0,1]\), let \(g(0+)\) and g(1-) exist, \(\int^{1}_{0}g(x)dx\leq 2,\) and suppose \(f'(a)f\prime''(b)- \frac{3}{2}f''(a)f''(b)<0\) for \(0\leq a<b\leq 1.\) The main result of the paper states that the convolution \(F(x)=\int^{\infty}_{-\infty}g(x-t)f(t) dt\) has continuous third derivative except possibly at \(x=1\), where at least F' is continuous, and \(F\prime''(x)/F'(x)-\frac{3}{2}(F''(x)/F'(x))^ 2<0\) for \(x\in (0,2)\).