One can show that the Iterated Function System consisting of transformations $S_1(x)=x, \;\;\ S_2(x)=\frac{1}{2}\;\;\; (x\in\mathbb{R})$ with constant probabilities $p_1=p_2=\frac{1}{2}$ is asymptotically stable with the Dirac measure $\delta_0$ as a unique invariant measure, that is for its Markov (dual) operator $P$ given by $Pf(x)=\frac{1}{2} \left(f(x) + f\left(\frac12 x\right)\right)\;\;\; (x\in \mathbb{R},\; f\in C_b(\mathbb{R})),$ where $C_b(\mathbb{R})$ stands for the set of all bounded continuous real-valued functions, we have $P^n f(x)\to \int_{\mathbb{R}} f(y)\,\delta_0(dy)=f(0)\;\;\; (f\in C_b(\mathbb{R})).$ My question is: Does the sequence $(P^n f)_{n\geq 1}$ converge uniformly to $f(0)$ on compact sets, for any $f\in C(\mathbb{R})$? I calculated that $P^n f(x)=\frac{1}{2^n}\sum_{k=0}^n {n\choose k} f\left(\frac{x}{2^k}\right)$
I suppose it does not, but I can't find an appropriate function. I have tried $x\mapsto (\sin x) /x$ (and 1 at $x=0$) but I do not know how to show it.