Turbulent Homeomorphisms and the Topological Snail

The paper states the turbulent fixed points theorem (NX(f) ≥ Tr(MT) and h_top(f) ≥ ln λ) for plane homeomorphisms determined by a finite sequence of top-displacements of the middle point and defines A, B, and the turbulence matrix MT in PSL2(Z) (Theorem 1) . It carefully develops the type A/B moves and their PSL2(Z) calculus via the topological snail and zip-relations , and even provides a combinatorial tree whose counts match the entries of MT and invokes “Smale’s method” to indicate existence of fixed points at branching nodes . However, it does not present a rigorous construction of a Markov partition or a factor map to the subshift with adjacency MT, nor a complete Nielsen-class separation argument yielding NX(f) ≥ Tr(MT). The only hint toward rectangles is an informal “thin rectangular box” remark for the colored snails . By contrast, the candidate solution gives a standard, self-contained proof: it encodes the isotopy by a graph map on a wedge of two arcs with transition matrix MT, realizes this with two Markov rectangles to obtain a symbolic factor and hence h_top(f) ≥ ln λ, and constructs Tr(MT) essential Nielsen fixed point classes from the diagonal ‘self-strips’. These are classical and sufficient steps missing in the paper’s exposition.