Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
6419288 | Journal of Mathematical Analysis and Applications | 2012 | 13 Pages |
Let f and g be elements of C(I) with xâI=[0,1]. We study the Ï-limit sets Ï(x,[f,g]) generated by alternating trajectories of the form γ(x,[f,g])={x,f(x),g(f(x)),f(g(f(x))),â¦}, as well as the sets Î([f,g])=âxâIÏ(x,[f,g]) and L([f,g])={Ï(x,[f,g]):xâI}. In particular, we show that(1)If g is constant on no interval JâI, then there exists a residual set SâC(I) so that the maps Î:C(I)ÃC(I)âK and L:C(I)ÃC(I)âKâ taking (f,g) to Î([f,g]) and L([f,g]), respectively, are both continuous at (f,g) whenever fâS.(2)The map Ï:IÃC(I)ÃC(I)âK taking (x,f,g) to Ï(x,[f,g]) is in the second class of Baire, and for any gâC(I) there exists a residual set TâIÃC(I) so that Ï is continuous at (x,f,g) whenever (x,f)âT.(3)If f is constant on no interval JâI, then there exists a residual set DâIÃC(I) so that Ï(x,[f,g])=Ï(x,gâf)âªÏ(f(x),fâg), where both Ï(x,gâf) and Ï(f(x),fâg) are adding machines of type â, whenever (x,g)âD.