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Equi-topological entropy curves for skew tent maps in the square

Buczolich, Zoltán and Keszthelyi, Gabriella (2017) Equi-topological entropy curves for skew tent maps in the square. MATHEMATICA SLOVACA. pp. 1-21. ISSN 0139-9918

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Abstract

We consider skew tent maps Tα,β(x) such that (α,β)∈[0,1]2 is the turning point of Tα,β, that is, Tα,β=βαx for 0≤x≤α and Tα,β(x)=β1−α(1−x) for α<x≤1. We denote by M−−=K(α,β) the kneading sequence of Tα,β and by h(α,β) its topological entropy. For a given kneading squence M−− we consider equi-kneading, (or equi-topological entropy, or isentrope) curves (α,φM−−(α)) such that K(α,φM−−(α))=M−−. To study the behavior of these curves an auxiliary function ΘM−−(α,β) is introduced. For this function ΘM−−(α,φM−−(α))=0, but it may happen that for some kneading sequences ΘM−−(α,β)=0 for some β<φM−−(α) with (α,β) still in the interesting region. Using ΘM−− we show that the curves (α,φM−−(α)) hit the diagonal {(β,β):0.5<β<1} almost perpendicularly if (β,β) is close to (1,1). Answering a question asked by M. Misiurewicz at a conference we show that these curves are not necessarily exactly orthogonal to the diagonal, for example for M−−=RLLRC the curve (α,φM−−(α)) is not orthogonal to the diagonal. On the other hand, for M−−=RLC it is. With different parametrization properties of equi-kneading maps for skew tent maps were considered by J.C. Marcuard, M. Misiurewicz and E. Visinescu.

Item Type: Article
Additional Information: közlésre elfogadva
Subjects: Q Science / természettudomány > QA Mathematics / matematika
SWORD Depositor: MTMT SWORD
Depositing User: MTMT SWORD
Date Deposited: 24 Feb 2017 13:55
Last Modified: 24 Feb 2017 13:55
URI: http://real.mtak.hu/id/eprint/49663

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