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### Open Access Journals

DCDS

By using the Carathéodory-Pesin structure(C-P structure), with
respect to arbitrary subset, the topological pressure and
topological entropy, introduced for a single continuous map, is
generalized to the cases of semigroup of continuous maps. Several of
their basic properties are provided.

DCDS

In the present paper, we are concerned with the Hausdorff dimension of certain sets arising in Engel continued fractions. In particular, the Hausdorff dimension of sets

$\big\{x ∈ [0,1): b_n(x) ≥ \phi (n)~i.m.~n ∈ \mathbb{N}\big\}\ \ \text{and}\ \ \big\{x ∈ [0,1): b_n(x) ≥ \phi(n),\ \forall n ≥ 1\big\}$ |

are completely determined, where

means infinitely many,

is the sequence of partial quotients of the Engel continued fraction expansion of

and

is a positive function defined on natural numbers.

$i.m.$ |

$\{b_n(x)\}_{n ≥ 1}$ |

$x$ |

$\phi$ |

DCDS

Let $f$ be a continuous map on a compact metric space. In this paper, under the hypothesis that $f$ satisfies the specification property, we prove that the set consisting of those points for which the Birkhoff ergodic average does not exist is either residual or empty.

DCDS

Let $f$ be a continuous transformation of a compact metric space $(X,d)$ and $\varphi$ any continuous function on $X$. In this paper, under the hypothesis that $f$ satisfies the specification property, we determine the topological entropy of the following sets:
$$K_{I}=\Big\{x\in X: A\big(\frac{1}{n}\sum_{i=0}^{n-1}\varphi(f^{i}(x))\big)=I\Big\},$$
where $I$ is a closed subinterval of $\mathbb{R}$ and $A(a_{n})$ denotes the set of accumulation points of the sequence $\{a_{n}\}_{n}$. Our result generalizes the classical result of Takens and Verbitskiy ( Ergod. Th. Dynam. Sys.,

*23*(2003), 317-348 ). As an application, we present another concise proof of the fact that the irregular set has full topological entropy if $f$ satisfies the specification property.
MBE

A classical deterministic SIR model is modified to take into account
of limited resources for diagnostic confirmation/medical isolation. We show that this modification
leads to four different scenarios (instead of three scenarios in comparison with the SIR model) for optimal isolation strategies, and obtain
analytic solutions for the optimal control problem that minimize the outbreak size under the assumption
of limited resources for isolation. These solutions and their corresponding optimal control policies are
derived explicitly in terms of initial conditions, model parameters and resources for isolation (such as the number of intensive care units).
With sufficient resources, the optimal control strategy is the normal Bang-Bang
control. However, with limited resources the optimal control strategy requires to switch to
time-variant isolation at an optimal rate proportional to the ratio of isolated cases over the entire infected population once the maximum capacity
is reached.

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