论文标题
分段的动力增加宫缩
Dynamics of piecewise increasing contractions
论文作者
论文摘要
令$ i_1 = [a_0，a_1），\ ldots，i_ {k} = [a_ {k-1}，a_k）$为间隔$ i = [0,1）$ of $ k $ subintervals的分区。令$ f：i \ to i $是一张地图，以使每个限制$ f | _ {i_i} $都是lipschitz的收缩。我们证明，任何$ f $都可以在上限很清晰的情况下最多允许$ k $周期性的轨道。我们还对分段线性$λ$ - 携带地图的动态感兴趣，其中$ 0 <λ<1 $。令$ b_1，\ ldots，b_k $为实数，让$f_λ：i \ to \ to \ mathbb {r} $是一个函数，使每个限制$f_λ| _ {i_i}（x）=λx +b_i $。在参数上的一般假设下f_λ（x）\ pmod {1} $在每个点等于周期性轨道，最多存在$ k $周期性轨道。此外，令$ \ mathfrak {e}^{（k）} $为$λ，a_1，\ ldots，a_ {k-1}，b_1，b_1，\ ldots，b_k $的特殊集合，我们证明了$ f $ n $ \ m m iesuly as a ploy as a e e e e e e e e e e e e e e e e e e e e e e} Hausdorff尺寸大或等于$ K $的度量集。
Let $I_1=[a_0,a_1),\ldots,I_{k}= [a_{k-1},a_k)$ be a partition of the interval $I=[0,1)$ into $k$ subintervals. Let $f:I\to I$ be a map such that each restriction $f|_{I_i}$ is an increasing Lipschitz contraction. We prove that any $f$ admits at most $k$ periodic orbits, where the upper bound is sharp. We are also interested in the dynamics of piecewise linear $λ$-affine maps, where $0<λ<1$. Let $b_1,\ldots,b_k$ be real numbers and let $F_λ: I\to \mathbb{R}$ be a function such that each restriction $F_λ|_{I_i}(x)=λx +b_i$. Under a generic assumption on the parameters $a_1,\ldots,a_{k-1},b_1,\ldots,b_k$, we prove that, up to a zero Hausdorff dimension set of slopes $λ$, the $ω$-limit set of the piecewise $λ$-affine maps $f_λ:x\in I \mapsto F_λ(x)\pmod{1}$ at every point equals a periodic orbit and there exist at most $k$ periodic orbits. Moreover, let $\mathfrak{E}^{(k)}$ be the exceptional set of parameters $λ,a_1,\ldots,a_{k-1},b_1,\ldots,b_k$ which define non-asymptotically periodic $f$, we prove that $\mathfrak{E}^{(k)}$ is a Lebesgue null measure set whose Hausdorff dimension is large or equal to $k$.
