Approximation for the invariant measures of Markov maps in \(\mathbb{R}^ d\) (Q1910187)

Let \(\varphi:I \to I\) be a Markov map from a \(d\)-dimensional domain \(I \subseteq R^d\) (not necessarily cube) into itself and let \(\pi\) be its Markov partition (i.e., \(\bigcup_{ V \in \pi} V=I\) up to a set of Lebesgue measure zero, and \(V \subset \varphi (U)\) if \(\varphi (U) \cap V \neq \emptyset)\). The discrete analogue of the Frobenius-Perron operator \(P_\varphi:L^1 (m) \to L^1 (m)\) is a matrix operator defined, for each integer \(r \geq 1\), as follows: for any \(q_r=\sum_{V \in \pi^r} q_V \mathbf{1}_V/m (V) \in L^1_r\) i.e. \(\int |q_r |dm=\sum_{V \in \pi^r} |q_V |< \infty\) \((m\) denotes the Lebesgue measure) where \(\pi^r=\bigvee^{r-1}_{j=0} \varphi^{-j} (\pi)\), we put \(Q_r q_r=\sum_{V \in \pi^r} a_V \mathbf{1}_V/m (V)\) with \(a_V=\sum_{U \in \pi^r} q_U q_{UV}\) and \(q_{UV}=m(U \cap \varphi^{- 2r} (V))/m (U)\). We discuss the problem of the convergence of the sequence \(\{q_{0r} \}\) of \(Q_r\)-invariant probabilistic densities (i.e., \(Q_r q_{0r}=q_{0r}\) and \(q_{0r} \geq 0\), \(|q_{0r} |=1)\) to a \(P_\varphi\)-invariant probabilistic density. This approximation problem is a modification of that one posed by \textit{S. M. Ulam} [A collection of mathematical problems, p. 75, (New York 1960; Zbl 0086.24101)]. The obtained result generalizes the results of \textit{T. Y. Li} [J. Approximation Theory 17, 177-186 (1976; Zbl 0357.41011)], the author [Z. Wahrscheinlichkeitstheorie verw. Geb. 59, 27-38 (1982; Zbl 0476.28010)], and \textit{A. Boyarsky} and \textit{Y. S. Lou} [J. Approximation Theory 65, No. 2, 231-244 (1991; Zbl 0737.41022)].