论文标题
Walters家族中的涉及内核和功能的双重潜力
The involution kernel and the dual potential for functions in the Walters family
论文作者
论文摘要
我们的符号：$ \ {0,1 \}^{\ Mathbb {z} - \ {0 \}} = \ {0,1 \}^\ Mathbb {n} \ times \ times \ times \ {0,1 \} $（y | x）=（...，y_2，y_1 | x_1，x_2，...）$，其中$（x_1，x_2，...） buignive地图$ \hatσ（...，y_2，y_1 | x_1，x_2，...）=（...，y_2，y_1，y_1，x_1 | x_2，...）$称为双侧移动，并在$ \ {0,1 \}给定的$ a：\ {0,1 \}^\ mathbb {n} =ω^+\ to \ mathbb {r} $我们在变量$ x $中表达$ a $，例如$ a（x）$。 以类似的方式，给定的$ b：\ {0,1 \}^\ mathbb {n} =ω^{ - } \ to \ mathbb {r} $我们在变量$ y $中表达$ b $，例如$ b（y）$。 最后，给定$ w：ω^{ - } \ timesω^{+} \ to \ mathbb {r} $，我们在变量$（y | x）$中表达$ w $，例如$ w（y | x）$。通过滥用符号，我们编写$ a（y | x）= a（x）$和$ b（y | x）= b（y）。$概率$μ_a$表示$ a的平衡概率： 给定连续电位$ a：ω^+\ to \ mathbb {r} $，我们说连续的潜在$ a^*：ω^{ - } \ to \ athbb {r} $是$ a $的双重潜在的潜力对于所有$（y | x）\ in \ {0,1 \}^{\ mathbb {z} - \ {0 \}} $ $$ a^*（y）= \ left [a \ circ \hatσ^{ - 1} + w \ circ \hatσ^{ - 1} - w \ right]（y | x）。 $$ 我们说$ w $是$ a $的互动内核。函数$ W $允许您在Ruelle Operator的主要特征功能的线性空间中定义光谱投影，以$ a $。 给定$ a $，我们描述了$ w $的明确表达方式和双势$ a^*$，用于P. Walters引入的功能系列中的$ a $。我们提出了$ a $对称和扭曲类型的条件。
Our notation: Points in $\{0,1\}^{\mathbb{Z}-\{0\}} =\{0,1\}^\mathbb{N}\times \{0,1\}^\mathbb{N}=Ω^{-} \times Ω^{+}$, are denoted by $( y|x) =(...,y_2,y_1|x_1,x_2,...)$, where $(x_1,x_2,...) \in \{0,1\}^\mathbb{N}$, and $(y_1,y_2,...) \in \{0,1\}^\mathbb{N}$. The bijective map $\hatσ(...,y_2,y_1|x_1,x_2,...)= (...,y_2,y_1,x_1|x_2,...)$ is called the bilateral shift and acts on $\{0,1\}^{\mathbb{Z}-\{0\}}$. Given $A: \{0,1\}^\mathbb{N}=Ω^+\to \mathbb{R}$ we express $A$ in the variable $x$, like $A(x)$. In a similar way, given $B: \{0,1\}^\mathbb{N}=Ω^{-}\to \mathbb{R}$ we express $B$ in the variable $y$, like $B(y)$. Finally, given $W: Ω^{-} \times Ω^{+}\to \mathbb{R}$, we express $W$ in the variable $(y|x)$, like $W(y|x)$. By abuse of notation we write $A(y|x)=A(x)$ and $B(y|x)=B(y).$ The probability $μ_A$ denotes the equilibrium probability for $A: \{0,1\}^\mathbb{N}\to \mathbb{R}$. Given a continuous potential $A: Ω^+\to \mathbb{R}$, we say that the continuous potential $A^*: Ω^{-}\to \mathbb{R}$ is the dual potential of $A$, if there exists a continuous $W: Ω^{-} \times Ω^{+}\to \mathbb{R}$, such that, for all $(y|x) \in \{0,1\}^{\mathbb{Z}-\{0\}}$ $$ A^* (y) = \left[ A \circ \hatσ^{-1} + W \circ \hatσ^{-1} - W \right] (y|x). $$ We say that $W$ is an involution kernel for $A$. The function $W$ allows you to define an spectral projection in the linear space of the main eigenfunction of the Ruelle operator for $A$. Given $A$, we describe explicit expressions for $W$ and the dual potential $A^*$, for $A$ in a family of functions introduced by P. Walters. We present conditions for $A$ to be symmetric and to be of twist type.
