    January  1996, 2(1): 111-120. doi: 10.3934/dcds.1996.2.111

## On chain continuity

 1 Mathematics Department, The City College, New York, N. Y. 10031, United States

Received  May 1995 Published  October 1995

A number of recent papers examine for a dynamical system $f: X \rightarrow X$ the concept of equicontinuity at a point. A point $x \in X$ is an equicontinuity point for $f$ if for every $\epsilon > 0$ there is a $\delta > 0$ so that the orbit of initial points $\delta$ close to $x$ remains at all times $\epsilon$ close to the corresponding points of the orbit of $x$, i.e. $d(x,x_0) < \delta$ implies $d(f^i(x),f^i(x_0)) \leq \epsilon$ for $i = 1,2,\ldots$. If we suppose that the errors occur not only at the initial point but at each iterate we obtain not the orbit of $x_0$ but a $\delta$-chain, a sequence $\{x_0,x_1,x_2,\ldots\}$ such that $d(f(x_i),x_{i+1}) \leq \delta$ for $i = 0,1,\ldots$. The point $x$ is called a chain continuity point for $f$ if for every $\epsilon > 0$ there is a $\delta > 0$ so that all $\delta$ chains beginning $\delta$ close to $x$ remain $\epsilon$ close to the points of the orbit of $x$, i.e. $d(x,x_0) < \delta$ and $d(f(x_i),x_{i+1}) \leq \delta$ imply $d(f^i(x),x_i) \leq \epsilon$ for $i = 1,2,\ldots$. In this note we characterize this property of chain continuity. Despite the strength of this property, there is a class of systems $(X,f)$ for which the chain continuity points form a residual subset of the space $X$. For a manifold $X$ this class includes a residual subset of the space of homeomorphisms on $X$.
Citation: Ethan Akin. On chain continuity. Discrete & Continuous Dynamical Systems - A, 1996, 2 (1) : 111-120. doi: 10.3934/dcds.1996.2.111
  Sushil Kumar Dey, Bibhas C. Giri. Coordination of a sustainable reverse supply chain with revenue sharing contract. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020165  Zonghong Cao, Jie Min. Selection and impact of decision mode of encroachment and retail service in a dual-channel supply chain. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020167  Jie Li, Xiangdong Ye, Tao Yu. Mean equicontinuity, complexity and applications. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 359-393. doi: 10.3934/dcds.2020167  Stefano Bianchini, Paolo Bonicatto. Forward untangling and applications to the uniqueness problem for the continuity equation. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020384  Giuseppina Guatteri, Federica Masiero. Stochastic maximum principle for problems with delay with dependence on the past through general measures. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020048

2019 Impact Factor: 1.338