Quasiregular analogues of critically finite rational functions with parabolic orbifold

Local dynamics of uniformly quasiregular mappings

“…The Lattès-type mappings introduced in [15], [16] are uqr analogues of the rational functions that are called critically finite with parabolic orbifold. They are obtained by semi-conjugating an expanding similarity by an automorphic map.…”

Rigidity in holomorphic and quasiregular dynamics

“…Nevertheless, it is a difficult task to write down explicit examples of uqr mappings having degree at least two. This was first done by Iwaniec and Martin [9], and subsequently other examples were given in [14], [17], [18]. In particular, Mayer's paper [17] exhibits the higher-dimensional Lattès maps.…”

Lattès-type mappings on compact manifolds

“…In 1997 and 1998, Mayer discovered an important family of examples of uniformly quasiregular mappings; see [10] and [11]. They are analogues of the rational functions that are called critically finite with parabolic orbifold.…”

“…[10]). He constructs chaotic uqr mappings acting on S 3 and analogues of power mappings with Julia set S 2 .…”

“…Power mapping. Let us first show how a 4-to-1 uqr mapping arises on S 3 via the Zorich mapping, the higher dimensional counterpart of the planar exponential function, as presented by Rickman [13, p. 15]) and later Mayer [10]. Denote the Zorich mapping by h Z : R 3 → R 3 \ {0}.…”