Cremer fixed points and small cycles

Let $\lambda= e^{2\pi i \alpha}$, $\alpha \in \mathbb{R}\setminus \mathbb{Q}$, and let $(p_n/q_n)$ denote the sequence of convergents to the regular continued fraction of $\alpha$. Let $f$ be a function holomorphic at the origin, with a power series of the form $f(z)= \lambda z+\sum _{n=2}^{\infty}a_nz^n$. We assume that for infinitely many $n$ we simultaneously have (i) $\log \log q_{n+1} \geq 3\log q_n$, (ii) the coefficients $a_{1+q_n}$ stay outside two small disks, and (iii) the series $f(z)$ is lacunary, with $a_j=0$ for $2+q_n\leq j \leq q_n^{1+q_n}-1$. We then prove that $f(z)$ has infinitely many periodic orbits in every neighborhood of the origin.