IFS完整概率的熵和变分原理
Entropy and Variational principles for holonomic probabilities of IFS
摘要：
与IFS相关联，可以考虑连续映射$\hatσ : [0,1]\times Σ\to [0,1]\times Σ$，由$\hatσ(x,w)=(τ_{X_{1}(w)}(x), σ(w))$定义为$Σ=\{0,1, ..., d-1\}^{\mathbb{N}}$，$σ: Σ\to Σ$由$σ(w_{1},w_{2},w_{3},...)=(w_{2},w_{3},w_{4}...)$给出，并且$X_{k} : Σ\to \{0,1, ..., n-1\}$是在坐标$k$上的投影。$ρ$加权系统$ρ\geq 0$是一个加权系统$([0,1], τ_{i}, u_{i})$，使得存在一个正有界函数$h : [0,1] \to \mathbb{R}$和概率$ν$在$[0,1]$上满足$ P_{u}(h)=ρh, \quad P_{u}^{*}(ν)=ρν$ .如果$ \int g \circ \hatσ d\hatν= \int g d\hatν, \forall g \in C([0,1])$，则$\hatν$关于$[0,1]\times Σ$的概率称为$\hatσ$的完整性。我们用${\cal H}$表示完整概率集。通过蜕变，$\hatν$对$[0,1]\times Σ$的完整概率自然地与$ρ$加权系统相关联。更准确地说，存在$[0,1]$上的概率$ν$和$[0,1]$上的概率$u_i, i\in\{0, 1,2,..,d-1\}$，即$P_{u}^*(ν)=ν$。我们考虑完整遍历概率。对于完整概率，我们定义熵。最后，我们对问题进行了分析：给定$ϕ\in \mathbb{B}^{+}$，求出最大压力问题$$p(ϕ)=$$的解
英文摘要：
Associated to a IFS one can consider a continuous map $\hatσ : [0,1]\times Σ\to [0,1]\times Σ$, defined by $\hatσ(x,w)=(τ_{X_{1}(w)}(x), σ(w))$ were $Σ=\{0,1, ..., d-1\}^{\mathbb{N}}$, $σ: Σ\to Σ$ is given by$σ(w_{1},w_{2},w_{3},...)=(w_{2},w_{3},w_{4}...)$ and $X_{k} : Σ\to \{0,1, ..., n-1\}$ is the projection on the coordinate $k$. A $ρ$-weighted system, $ρ\geq 0$, is a weighted system $([0,1], τ_{i}, u_{i})$ such that there exists a positive bounded function $h : [0,1] \to \mathbb{R}$ and probability $ν$ on $[0,1]$ satisfying $ P_{u}(h)=ρh, \quad P_{u}^{*}(ν)=ρν$. A probability $\hatν$ on $[0,1]\times Σ$ is called holonomic for $\hatσ$ if $ \int g \circ \hatσ d\hatν= \int g d\hatν, \forall g \in C([0,1])$. We denote the set of holonomic probabilities by ${\cal H}$. Via disintegration, holonomic probabilities $\hatν$ on $[0,1]\times Σ$ are naturally associated to a $ρ$-weighted system. More precisely, there exist a probability $ν$ on $[0,1]$ and $u_i, i\in\{0, 1,2,..,d-1\}$ on $[0,1]$, such that is $P_{u}^*(ν)=ν$. We consider holonomic ergodic probabilities. For a holonomic probability we define entropy. Finally, we analyze the problem: given $ϕ\in \mathbb{B}^{+}$, find the solution of the maximization pressure problem $$p(ϕ)=$$
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