A Monotonicity Conjecture for Real Cubic Maps . 1
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چکیده
Section 2 sets the stage by describing the parameter triangle T for real cubic maps, either of shape + + or of shape + , and by describing basic properties of topological entropy. Section 3 describes the monotonicity problem for the topological entropy function, and states the Monotonicity Conjecture. Section 4 describes the family of stunted sawtooth maps, and proves the analogous conjecture for this family. Sections 5 begins to relate these two families by describing the `bone' structure in the parameter triangle. By de nition, a bone B (o) in the triangle T is the set of parameter points v such that a speci ed critical point (left or right) of the associated bimodal map belongs to a periodic orbit with speci ed order type o . (Compare [MaT].) It is conjectured that every bone is a simple connected arc in T . Although we cannot prove either of these conjectures for cubic maps, we do show that Generic Hyperbolicity ) Connected Bone Conjecture ) Monotonicity Conjecture
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