论文标题
圆形图的随机扰动的热力学形式主义和中央限制定理
The thermodynamic formalism and central limit theorem for stochastic perturbations of circle maps with a break
论文作者
论文摘要
令$ t \ in c^{2+ \ varepsilon}（s^{1} \ setMinus \ {x_ {x_ {b} \}），\，\，\ varepsilon> 0，$ be be be be be a in eartientation Circe homeMorphism andomorphism andomorphism an $ρ_t= [k_ {1}，k_ {2}，..，k_ {m}，1,1，...]，\，\，\，m \ geq1 $和单个断点$ x_ {b} $。我们考虑随机序列 $ \ OVERLINE {z} _ {n+1}（z_0，σ）= t（\ overline {z} _ {n}）+σ取项_ {n+1}，\，\，\，\，\，\，\，{z} _ {0} _ {0} $ \ {ξ_{n}，\，n = 1,2，... \} $是一个真实有价值的独立平均零随机变量的序列，具有可比大小的零随机变量，$σ> 0 $是一个小参数。使用重新归一化组技术De la Llave等。被证明是一二姆的随机扰动。间隔映射一个中心极限定理（CLT）和收敛速率。在本文中，我们通过使用Dzhalilov等人最近构建的热力学形式主义来将其结果扩展到具有断裂点的同态形态。这种形式主义和动态分区$ p_n（t，x_b）$由断裂点确定的，遵循Vul等人的工作，可以为S^1 $中的任何$ z \建立符号动态，并定义一个转移操作员，其领先的EigenValue用于限制Lyapunov的功能。对于特殊序列$ \ {n_m \}，m \ to \ infty $，任何$ z_k = t^kz_0 $的barycentric系数不与$ x_b $的轨道相交的范围是普遍限制在$ p_ {n_m}（t，x__b）$中的相应间隔中的。 $ \ Overline {z} _ {n}（Z_0，σ）$的Taylor扩展在$ \ {ξ_i\} $中的$将分解成$ t^n（z_0）$，一个线性化的有效噪声和更高级别的噪声和更高级别的噪声和更高的顺序项。但是，这仅在某些社区中才有可能，$ a_k^{n_m} $ of点的点$ t^k z_0 $不包含$ t^{q_ {n_m}} $的断点，带有$ q_ {n} $ $ t $的第一个返回时间。证明线性化过程的CLT最终导致了我们扩展了De la Llave等人的结果。
Let $T\in C^{2+\varepsilon}(S^{1}\setminus\{x_{b}\}),\,\,\varepsilon>0,$ be an orientation preserving circle homeomorphism with rotation number $ρ_T=[k_{1},k_{2},..,k_{m},1,1,...],\,\,m\geq1$, and a single break point $x_{b}$. We consider the stochastic sequence $ \overline{z}_{n+1}(z_0,σ) = T(\overline{z}_{n}) + σξ_{n+1},\,\overline{z}_{0}:=z_0\in S^1$, where $\{ξ_{n},\,n=1,2,...\}$ is a sequence of real valued independent mean zero random variables of comparable sizes, and $σ> 0$ is a small parameter. Using the renormalization group technique de la Llave et al. proved for stochastic perturbations of one-dim. interval maps a central limit theorem (CLT) and the rate of convergence. In the present paper we extend their results to circle homeomorphisms with a break point by using the thermodynamic formalism constructed recently by Dzhalilov et al.. for such maps. This formalism and the dynamical partition $P_n(T,x_b)$ determined by the break point allows us, following the work of Vul et al., to establish a symbolic dynamics for any $z\in S^1$ and to define a transfer operator whose leading eigenvalue is used to bound the Lyapunov function. For a special sequence $\{n_m\}, m\to\infty$, the barycentric coefficient of any $z_k=T^kz_0$ not intersecting the orbit of $x_b$ is universally bounded in the corresponding interval in $P_{n_m}(T,x_b)$. A Taylor expansion of $ \overline{z}_{n}(z_0,σ)$ in $\{ξ_i\}$ leads to the decomposition into the term $T^n(z_0)$, a linearized effective noise and higher order terms in $\{ξ_i\}$. This is possible however only in certain neighbourhoods $A_k^{n_m}$ of the points $T^k z_0$ not containing break points of $T^{q_{n_m}}$, with $q_{n}$ the first return times of $T$. Proving the CLT for the linearized process leads finally to the proof of our extension of results of de la Llave et al..
