Equi-topological entropy curves for skew tent maps in the square
Equi-topological entropy curves for skew tent maps in the square
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正方形倾斜帐篷图的等拓扑熵曲线
DOI：
10.1515/ms-2017-0072
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发表时间：
2015
期刊：
影响因子：
--
通讯作者：
Gabriella Keszthelyi
中科院分区：
文献类型：
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作者：
Z. Buczolich;Gabriella Keszthelyi
我们考虑斜帐篷映射Tα，β(X)使得(α，β)∈[0，1]2是TTα，β的转折点，即Tα，β=βα$\Begin{≤}{}\FRAC{{\α，β}}{{\Alpha}}\End{α，β=βα}$x对于0≤x≤α，Tα，β(X)=β1−α$\Begin{−}{}\FRAC{{\α}}{1-{\Alpha}}\End{数组}$(1 BEAT X)；X≤1.我们用M=K(α，β)表示TTα，β的捏合序列，用h(α，β)表示它的拓扑熵。对于给定的捏合序列M，我们考虑等捏合(或等拓扑熵或等熵)曲线(α，φM(α))使得K(α，φM(α))=M。为了研究这些曲线的行为，引入了辅助函数ΘM(α，β)。对于这个函数，ΘM(α，φM(α))=0，但对于某些捏合序列，ΘM(α，β)=0，对于某些β<；φM(α)，(α，β)仍然在单位平方的动态有趣的四分之一。利用ΘM，我们证明了当(α，φ)接近(1，1)时，曲线(β，β)几乎垂直于对角线{(β，β)：0.5lt；β<；1}。在回答M.Misiurewicz在一次会议上提出的问题时，我们证明了这些曲线不一定与对角线完全正交，例如，对于M=RLLRC，曲线(α，φM(α))不与对角线正交。另一方面，对于M=RLC来说，它是。J.C.Marcuard，M.Misiurewicz和E.Visinescu考虑了斜帐篷映射等捏映射的不同参数化性质。
Abstract We consider skew tent maps Tα, β(x) such that (α,β)∈[0,1]2 is the turning point of TTα, β, that is, Tα, β = βα $\begin{array}{} \frac{{\beta}}{{\alpha}} \end{array} $x for 0≤ x ≤ α and Tα, β(x) = β1−α $\begin{array}{} \frac{{\beta}}{1- {\alpha}} \end{array} $ (1−x) for α < x ≤ 1. We denote by M = K(α,β) the kneading sequence of TTα, 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 dynamically interesting quarter of the unit square. 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.