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Let $A$ be a limsup random fractal with indices $\gamma_1, ~\gamma_2 ~$and $\delta$ on $[0,1]^d$. We determine the hitting probability $\mathbb{P}(A\cap G)$ for any analytic set $G$ with the condition $(\star)$$\colon$ $\dim_{\rm H}(G)>\gamma_2+\delta$, where $\dim_{\rm H}$ denotes the Hausdorff dimension. This extends the correspondence of Khoshnevisan, Peres and Xiao [10] by relaxing the condition that the probability $P_n$ of choosing each dyadic hyper-cube is homogeneous and $\lim\limits_{n\to\infty}\frac{\log_2P_n}{n}$ exists. We also present some counterexamples to show the Hausdorff dimension in condition $(\star)$ can not be replaced by the packing dimension., Comment: 12 pages

The goal of this paper is to define and investigate those topological pressures, which is an extension of topological entropy presented by Feng and Huang [13], of continuous transformations. This study reveals the similarity between many known results of topological pressure. More precisely, the investigation of the variational principle is given and related propositions are also described. That is, this paper defines the measure theoretic pressure $P_{\mu}(T,f)$ for any $\mu\in{\mathcal M(X)}$, and shows that $P_B(T,f,K)=\sup\bigr\{P_{\mu}(T,f):{\mu}\in{\mathcalM(X)},{\mu}(K)=1\bigr\}$, where $K\subseteq X$ is a non-empty compact subset and $P_B(T,f,K)$ is the Bowen topological pressure on $K$. Furthermore, if $Z\subseteq X$ is an analytic subset, then $P_B(T,f,Z)=\sup\bigr\{P_B(T,f,K):K\subseteq Z\ \text{is compact}\bigr\}$. However, this analysis relies on more techniques of ergodic theory and topological dynamics., Comment: 15 pages. arXiv admin note: substantial text overlap with arXiv:1012.1103, arXiv:1111.7121 by other authors

This paper defines the pressure for asymptotically subadditive potentials under a mistake function, including the measuretheoretical and the topological versions. Using the advanced techniques of ergodic theory and topological dynamics, we reveals a variational principle for the new defined topological pressure without any additional conditions on the potentials and the compact metric space., Comment: 13pages