On the support of measures of large entropy for polynomial-like maps

IF 1.4 3区 数学 Q1 MATHEMATICS
Sardor Bazarbaev, Fabrizio Bianchi, Karim Rakhimov
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引用次数: 0

Abstract

Let f be a polynomial-like map with dominant topological degree \(d_t\ge 2\) and let \(d_{k-1}<d_t\) be its dynamical degree of order \(k-1\). We show that every ergodic measure whose measure-theoretic entropy is strictly larger than \(\log \sqrt{d_{k-1} d_t}\) is supported on the Julia set, i.e., the support of the unique measure of maximal entropy \(\mu \). The proof is based on the exponential speed of convergence of the measures\(d_t^{-n}(f^n)^*\delta _a\) towards \(\mu \), which is valid for a generic point a and with a controlled error bound depending on a. Our proof also gives a new proof of the same statement in the setting of endomorphisms of \(\mathbb P^k(\mathbb C)\) – a result due to de Thélin and Dinh – which does not rely on the existence of a Green current.

类多项式映射的大熵测度支持
设f为一个具有优势拓扑度\(d_t\ge 2\)的类多项式映射,设\(d_{k-1}<d_t\)为其动态阶次\(k-1\)。我们证明了在Julia集合上支持所有测度理论熵严格大于\(\log \sqrt{d_{k-1} d_t}\)的遍历测度,即最大熵的唯一测度\(\mu \)的支持。该证明是基于测量\(d_t^{-n}(f^n)^*\delta _a\)到\(\mu \)的指数收敛速度,它对一般点a有效,并且具有依赖于a的可控误差界。我们的证明还在\(\mathbb P^k(\mathbb C)\)的自同态设置中给出了相同陈述的新证明-这是由于de th和Dinh的结果-它不依赖于格林电流的存在。
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来源期刊
Analysis and Mathematical Physics
Analysis and Mathematical Physics MATHEMATICS, APPLIED-MATHEMATICS
CiteScore
2.70
自引率
0.00%
发文量
122
期刊介绍: Analysis and Mathematical Physics (AMP) publishes current research results as well as selected high-quality survey articles in real, complex, harmonic; and geometric analysis originating and or having applications in mathematical physics. The journal promotes dialog among specialists in these areas.
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