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Let$f:V$→$R$be a function defined on a subset$V$of$R$^{$n$}×$R$^{$d$}let$⃜:x→$inf{$f(x t)$;$t$such that$(x t)∈V}$denote the$shadow$of$f$and let$Φ$=${(x t)∈V; f(x t)=⃜(x)}$This paper deals with the characterization of some properties of ⃜ in terms of the infinitesimal behavior of$f$near points ζ∈$Φ$proving in particular a conjecture of J M Trépreau concerning the case$d$=1 Characterizations of this type are provided for the convexity the subharmonicity or the$C$^{1 1}regularity of ⃜ in the interior of$I={x∈$$R$^{n};ε$R$^{d}$(x t)∈V}$and in the$C$^{1 1}case an expression for$D$^{2}⃜ is given To some extent an answer is given to the following question: which convex function ⃜:$I$→$R$$I$interval ϒ$R$(resp which function √:$I$→$R$of class$C$^{1 1}) is the shadow of a$C$^{2}function$f:I$×$R→R$?

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We answer Hubbard's question on determining the Thurston equivalence class of “twisted rabbits”, i.e. composita of the “rabbit” polynomial with$n$th powers of the Dehn twists about its ears. The answer is expressed in terms of the 4-adic expansion of$n$. We also answer the equivalent question for the other two families of degree-2 topological polynomials with three post-critical points. In the process, we rephrase the questions in group-theoretical language, in terms of wreath recursions.