Milnor’s Conjecture on Monotonicity of Topological Entropy: results and questions∗

نویسنده

  • Sebastian van Strien
چکیده

Theorem (Milnor and Thurston [MT77], see also [MT88]). The function C → R which associates to a mapping g ∈ C its topological entropy htop(g) is continuous. Here C stands for C maps of the interval with b non-degenerate critical points (non-degenerate means second derivative non-zero). This theorem relies on a result of Misiurewicz and Szlenk, see [MS77, MS80] who had previously shown that γ(f) := exp(htop(f)) is equal to the growth rate of the number of laps (i.e., intervals of monotonicity) of f. The crucial new ingredient in the proof of Milnor and Thurson’s theorem is a formula which shows that γ(f) is also the zero of a certain meromorphic function.

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تاریخ انتشار 2012