Certain parity functions were used by Terras in his study of the Collatz map. One can combine these into a single function, call it gamma, understood as an isometry (and hence measure preserving bijection) of the ring Z_2 of dyadic integers. The map gamma conjugates the Collatz map T (extended to Z_2) to the one-sided shift map on Z_2 and, thus, T is ergodic on Z_2. Lagarias related properties of gamma to the Collatz conjecture and other conjectures associated to it. For instance, the Collatz Conjecture is equivalent to the claim that gamma(Z^+) is a subset of 1/3*Z, while the conjecture that all orbits of the Collatz map on Z are eventually periodic is equivalent to the claim that gamma(Z_2 intersected with Q) = Z_2 intersected with Q. Bernstein and Lagarias provided some results on the cycle structure of gamma and, based on these results and empirical observations, they stated several conjectures.
We revisit the study of gamma, but think of it as a binary tree automorphism. In fact, we start with a more general approach. Namely, if alpha_0 and alpha_1 are two binary tree automorphisms and T is defined by cases, as \alpha_0(u/2) for even u and alpha_1((u-1)/2) for odd u in Z_2, that is, for all w, T(0w)=alpha_0(w) and T(1w)=alpha_1(w), we define gamma as the unique tree automorphism that conjugates T to the shift map and preserves parity. We prove that, if both alpha_0 and alpha_1 are finite state automorphisms (which they are in case of T), then gamma(Z_2 intersected with Q) contains Z_2 intersected with Q. Moreover, if the minimal self-similar group generated by alpha_0 and alpha_1 is contracting then we have equality, that is, gamma(Z_2 intersected with Q) = Z_2 intersected with Q.
In the concrete case when gamma is the conjugator of the Collatz map, by using the tools and the language of tree automorphisms we reprove the results of Bernstein and Lagarias and provide some further insights into gamma. We end with some new empirical data and open questions. In particular, we provi

A Fibonacci map is a piecewise defined map of a subset of an interval I onto I with a unique critical point of order d whose orbit undergoes nearest returns at Fibonacci times. It has been shown by Bruin, Keller, Nowicki and van Strien that such maps exhibit wild attractors: Cantor sets of zero Lebesgue measure whose basin of attraction is meager but has positive Lebesgue measure. We will discuss real renormalization, and a trichotomy for Fibonacci maps, similar to the Avila-Lyubich trichotomy for Feigenbaum Julia sets, which, in particular, allows us to show that Fibonacci maps admit wild attractors for d=5.1, and do not for d=3.9 (and, conjecturally, for 2 < d < = 3.9).

     The Iterated Monodromy group IMG(z^2+i), together with the Basilica group IMG(z^2-1) are in the corner of the theory of self-similar groups developed in the works of Grigorchuk, Bartholdi, Nekrashevych and other researches. It appeared for the first time in their joint article "From Fractal Groups to Fractal Sets" (2002) as an example of a group with the limit space homeomorphic to the Julia set of the polynomial z^2+i and topologically being a dendrite. Later it was discovered that it has an intermediate growth between polynomial and exponential (K-U.Bux and R.Perez), has a finite L-presentation, and that it shares many other properties similar to the Grigorchuk group. Spectral properties of the group were studied by Grigorchuk, Sunik and Savchuk and are not completely known yet. In the study of groups acting on rooted trees an important property is the Congruence Subgroup Property (CSP). If it holds then the closure of the group in the group Aut(T) of automorphisms of the tree coincides with the profinite completion. Absence of CSP makes the study of the profinite completion more difficult.
     In my talk I will show that IMG(z^2+i) does not satisfy CSP. For doing this I will inspect some important properties of the subgroup lattice of IMG(z^2+i).