Inverse pressure for finitely generated semigroups (Q2145632)

The authors introduce and study an inverse topological pressure \(P^{-}\) and its capacities for multi-potentials with respect to the dynamics of finitely generated semigroups \(G\) of non-invertible maps on arbitrary sets \(Y\) of a compact metric space \(X\). Section 1 is introductory. In Section 2, the authors introduce the notion of inverse pressure \(P^{-}(\Phi,Y,G_1)\) for a multipotential \(\Phi=(\Phi_1,\dots,\Phi_m)\in C(X,\mathbb{R}^m)\) with respect to a semigroup \(G\) generated by a finite set \(G_1\) of endomorphisms of a compact metric space \(X\), on a subset \(Y\subset X\). The following important result is established there. {Theorem}. Assume that \(f_i:M\to M\) are \(C^1\)-differentiable maps on a differentiable manifold \(M\), and there exists a compact set \(A\) such that \(f_i(A)\subset A\), for \(1\le i\le m\). Further, assume that there exits \(d\ge 1\) such that any point in \(A\) has at most \(d\) \(f_i\)-preimages for each \(i\in \{1,\dots,m\}\). Then, if \(G_1=\{\mathrm{id}_A,f_1|_{A},\dots,f_m|_{A}\}\), it follows that \(h^{-}(G_1)< \infty\), where \(h^{-}(G_1)\) stands for the inverse entropy on \(X\) with respect to \(G_1\). Finally, in Section 4 the authors introduce the notion of lower and upper local inverse entropies for probability measures, relative to the action of a semigroup \(G\). The following result is obtained. {Theorem}. Let \(\mu\) be a probability measure on \(X\), and \(Y\subset X\) be a Borel set with \(\mu(Y)> 0\). Consider a semigroup of self-maps on \(X\) generated by a finite set \(G_1\), and assume that \(\alpha> 0\) is such that \(h_l^{-}(x,\mu,G_1)\ge \alpha\) for each \(x\in Y\). Then, \(h^{-}(Y,G_1)\ge \alpha\).