Topological Conjugate Piecewise Linear Unimodal Mappings of an Interval Into Itself

Topological conjugation of one dimensional maps

“…Proposition (Proposition 3.5, also Lemma 1 in [32]). Let f, g : [0, 1] → [0, 1] be piecewise linear maps, which are topologically conjugated via increasing piecewise linear homeomorphism h. If f (0) = 0, then g(0) = 0 and f ′ (0) = g ′ (0).…”

“…Theorem (Theorem 11.3 also Theorem 2 in [32]). For an arbitrary v ∈ (0, 1) and an increasing piecewise linear maps g : [0, v] → [0, 1] such that g(0) = 0, g(v) = 1 and g ′ (0) = 2 there exists and the unique its continuation g : [0, 1] → [0, 1], which is topologically conjugated to f of the form (2.2) via piecewise linear homeomorphism.…”

“…Theorem (Theorem 11.1, also Theorem 1 in [32]). Let the maps f , which is given by (2. for some 0 α < β 1, then h is piecewise linear.…”

Classification of commutative pairs of surjective maps of interval, one of which is unimodal

“…For example, for the maps g and ψ from Fig. 7 we have that P = {(0, 0); (1,3); (2,6); (3,9); (4, 12)} and Q = {(0, 0); (4, 2); (6,3); (8,4); (10, 5); (12, 6)}. Thus, we cave naturally come to the classification of the commutative piecewise linear pairs g and ψ, where g is unimodal surjective such that g(0) = g(1) = 0.…”

“…By (3.8), rewrite (3.5) as 3 (3.9) Plug (3.9) into (3.8) and, after simplification, get 3 (3.10) Substitute (3.9) and (3.10) into (3.6):…”