论文标题
沿渐近线性独立序列的混合和刚性
Mixing and rigidity along asymptotically linearly independent sequences
论文作者
论文摘要
我们利用高斯度量保存系统来证明Lebesgue测量保留转换的存在和通用性$ t：[0,1] \ rightarrow [0,1] $，这些[0,1] $沿渐近线性独立序列的家庭表现出混合和刚性行为。令$λ_1，...，λ_n\在[0,1] $中，让$ ϕ_1，...，...，ϕ_n：\ mathbb n \ rightArrow \ rightBb z $在线性上是差异的独立的（即任何$（对于任何$（a_1，...，... $ \ lim_ {k \ rightarrow \ infty} | \ sum_ {j = 1}^na_j ϕ_j（k）| = \ infty $）。然后，可逆的lebesgue等级保存转换$ t：[0,1] \ rightarrow [0,1] $，其中存在一个序列$（n_k）_ {k \ in \ mathbb n} $ in \ mathbb n} $ in $ \ mathbb n $ in $ \ mathbb n $ with $ \ mathbb n lim $ \ lim_ \ lim_ \ lim_ { } b）=（1-λ_j）μ（a \ cap b）+λ_jμ（a）μ（b），对于任何可测量的$ a，b \ subseteq [0,1] $和任何$ j \ in \ in \ {1，...，...，n \} $的$$。该结果是由于A. M.Stëpin（参见“通用动力系统的光谱特性”中的定理2）以及由于V. Bergelson，S。Kasjan和M.Lemańczyk引起的结果的改进（请参阅“单一的操作员和同性恋者和IDEMPOTETTERT ULSTRAFERTERS”）。
We utilize Gaussian measure preserving systems to prove the existence and genericity of Lebesgue measure preserving transformations $T:[0,1]\rightarrow [0,1]$ which exhibit both mixing and rigidity behavior along families of asymptotically linearly independent sequences. Let $λ_1,...,λ_N\in[0,1]$ and let $ϕ_1,...,ϕ_N:\mathbb N\rightarrow\mathbb Z$ be asymptotically linearly independent (i.e. for any $(a_1,...,a_N)\in\mathbb Z^N\setminus\{\vec 0\}$, $\lim_{k\rightarrow\infty}|\sum_{j=1}^Na_jϕ_j(k)|=\infty$). Then the class of invertible Lebesgue measure preserving transformations $T:[0,1]\rightarrow[0,1]$ for which there exists a sequence $(n_k)_{k\in\mathbb N}$ in $\mathbb N$ with $$\lim_{k\rightarrow\infty}μ(A\cap T^{-ϕ_j(n_k) }B)= (1-λ_j)μ(A\cap B)+λ_jμ(A)μ(B),$$ for any measurable $A,B\subseteq [0,1]$ and any $j\in\{1,...,N\}$, is generic. This result is a refinement of a result due to A. M. Stëpin (see Theorem 2 in "Spectral properties of generic dynamical systems") and a generalization of a result due to V. Bergelson, S. Kasjan, and M. Lemańczyk (see Corollary F in "Polynomial actions of unitary operators and idempotent ultrafilters").
