A counterexample to the multifractal rigidity conjecture of piecewise linear Markov maps of the interval
A counterexample to the multifractal rigidity conjecture of piecewise linear Markov maps of the interval
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区间分段线性马尔可夫图多重分形刚性猜想的反例
DOI：
10.1080/14689367.2016.1153606
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发表时间：
2016
期刊：
影响因子：
--
通讯作者：
Katsukuni Nakagawa
中科院分区：
文献类型：
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作者：
Katsukuni Nakagawa
重分形刚性是指两个重分形谱的重合意味着给定的两个动力系统之间的联系更强的现象。对于定义在分段线性马尔可夫映射的排斥子上的Bernoulli测度的维谱，Barreira，Pesin和Schmeling证明了如果排斥子是由两个符号的全移位建模的，则刚性成立。在有三个或更多符号的情况下，这种僵化是否成立是一个悬而未决的问题。我们通过构造一个反例给出了这个问题的否定答案，并证明了我们的反例在三个符号的情况下是唯一的。
ABSTRACT A multifractal rigidity is the phenomena that the coincidence with the two multifractal spectra implies more strong relation between the given two dynamical systems. For the dimension spectrum of a Bernoulli measure defined on a repeller of a piecewise linear Markov map, Barreira, Pesin and Schmeling proved that a rigidity holds if the repeller is modelled by the full shift of two symbols. It was an open problem whether this rigidity holds in the case of three or more symbols. We give a negative answer to this problem by constructing a counter-example and we show that our counter-example is the unique one in the case of three symbols.