论文标题
诱导非自治动力学系统的动力学
Induced dynamics of non-autonomous dynamical systems
论文作者
论文摘要
令$ f_ {0，\ infty} = \ {f_n \} _ {n = 0}^{\ infty} $是紧凑型公制空间$ x $上的连续自动图的顺序。非自主动力学系统$（x，f_ {0，\ infty}）$诱导set-valued system $（\ nathcal {k} {k}（x），\ bar {f} _ {0，\ infty}）$ and Fuzzified System $ $（\ Mathcal {f}（x），\ tilde {f} _ {0，\ infty}）$。我们证明，在某些自然条件下，$（x，f_ {0，\ infty}）$的积极拓扑熵意味着$（\ Mathcal {k}（x）（x），\ bar {f} _ {0，\ infty}）$的无限熵$（\ Mathcal {f}（x），\ tilde {f} _ {0，\ infty}）$; $（s^1，f_ {0，\ infty}）$的零熵表示$（\ Mathcal {K}（s^1）的某些不变子系统的零熵$（\ Mathcal {f}（s^1），\ tilde {f} _ {0，\ infty}）$。我们确认$（\ MATHCAL {K}（i），\ bar {f}）$和$（\ Mathcal {f}（i），\ tilde {f}）$对于任何瞬时间隔$ f $均具有无限熵。相反，我们构建了一个传递的非自主系统$（i，f_ {0，\ infty}）$，使得两个$（\ nathcal {k}（k}（i），\ bar {f} _ {0，\ infty}）零熵。我们获得$（\ Mathcal {k}（x），\ bar {f} _ {0，\ infty}）$是链条链链薄弱，仅当且仅当$（\ nathcal {\ mathcal {f}^1（x）（x），\ tilde {f} _ {f} _ {f} _ {0，\ fistty} $ so（so）和chine和chine和cains y是。 $（x，f_ {0，\ infty}）$，$（\ MATHCAL {k}（x）（x），\ bar {f} _ {0，\ infty}）$多 - $ \ $ \ mathscr {f} $ - 敏感性）$（x，f_ {0，\ infty}）$ $（\ MATHCAL {F}^1（x），\ tilde {f} _ {0，\ infty}）$是等效的，其中$（\ Mathcal {f}^1（x），\ tilde {f} _ {f} _ {0，\ infty}）$是允许的正常fuzz pifife。
Let $f_{0,\infty}=\{f_n\}_{n=0}^{\infty}$ be a sequence of continuous self-maps on a compact metric space $X$. The non-autonomous dynamical system $(X,f_{0,\infty})$ induces the set-valued system $(\mathcal{K}(X), \bar{f}_{0,\infty})$ and the fuzzified system $(\mathcal{F}(X),\tilde{f}_{0,\infty})$. We prove that under some natural conditions, positive topological entropy of $(X,f_{0,\infty})$ implies infinite entropy of $(\mathcal{K}(X),\bar{f}_{0,\infty})$ and $(\mathcal{F}(X),\tilde{f}_{0,\infty})$, respectively; and zero entropy of $(S^1,f_{0,\infty})$ implies zero entropy of some invariant subsystems of $(\mathcal{K}(S^1),\bar{f}_{0,\infty})$ and $(\mathcal{F}(S^1),\tilde{f}_{0,\infty})$, respectively. We confirm that $(\mathcal{K}(I), \bar{f})$ and $(\mathcal{F}(I), \tilde{f})$ have infinite entropy for any transitive interval map $f$. In contrast, we construct a transitive non-autonomous system $(I, f_{0,\infty})$ such that both $(\mathcal{K}(I), \bar{f}_{0,\infty})$ and $(\mathcal{F}(I), \tilde{f}_{0,\infty})$ have zero entropy. We obtain that $(\mathcal{K}(X),\bar{f}_{0,\infty})$ is chain weakly mixing of all orders if and only if $(\mathcal{F}^1(X),\tilde{f}_{0,\infty})$ is so, and chain mixing (resp. $h$-shadowing and multi-$\mathscr{F}$-sensitivity) among $(X,f_{0,\infty})$, $(\mathcal{K}(X),\bar{f}_{0,\infty})$ and $(\mathcal{F}^1(X),\tilde{f}_{0,\infty})$ are equivalent, where $(\mathcal{F}^1(X),\tilde{f}_{0,\infty})$ is the induced normal fuzzification.
