Kupka-Smale extensions of Cantor set homeomorphisms without sources or sinks (Q1336613)

The main theorem of this paper is as follows: if \(f\) is a block-permuting homeomorphism of the Cantor set without periodic orbits, then there exists a \(C^ 1\) Kupka-Smale diffeomorphism \(F\) of the 2-disk without periodic sources or sinks, and an invariant Cantor set \(\Lambda\) such that the restriction of \(F\) to \(\Lambda\) is conjugate to \(f\). The notion of block-permuting homeomorphisms of a Cantor set is due to the authors. The class of block-permuting homeomorphisms includes all \(k\)-symbol adding machines and all finite products of them. The authors conjecture that every \(C^ 1\) Kupka-Smale diffeomorphism of the 2-disk without sources or sinks has an invariant Cantor set which supports a 2-symbol adding machine.