Weighted Entropy of a Flow on Non-compact Sets

Let \((X_i,\phi _i), i=1,2,\ldots ,k\) be continuous flows on compact metric spaces, and for each \(1\le i\le k-1\), \((X_{i+1},\phi _{i+1})\) be a factor of \((X_i,\phi _i)\). Let \(\mathbf{a}=(a_1,a_2,\ldots ,a_k)\in {\mathbb {R}}^k\) with \(a_1>0\) and \(a_i\ge 0\) for \(2\le i\le k\). Based on the theory of Carathéodory structure, this paper introduce the \(\mathbf{a}\)-weighted topological entropy of a flow on non-compact sets and the \(\mathbf{a}\)-weighted measure-theoretic entropy of a flow. We establish the variational principle and also investigate the relationship between the \(\mathbf{a}\)-weighted entropy and the classical \(\mathbf{a}\)-weighted entropy of time one map.