Measure preserving extensions and minimal wandering rates
This paper is a continuation of previous works, in particular of J. Aaronson, M. Thaler and the author, on the notion of wandering rate for measure preserving transformations in infinite-measure ergodic theory. For a conservative ergodic measure preserving (C.E.M.P.) dynamical system \(\mathsf{T} = (X, \mathcal A, \mu, T)\) on a \(\sigma\)-finite measure space, the ``wandering rate'' of a set \(A\) in \(\mathcal A\) is the sequence \(w_N(A):=\mu(\bigcup_{n=0}^{N-1} T^{-n} A), \;N \geq 1\). A set \(Y\) in \(\mathcal A\) with positive finite measure is said to have minimal wandering rate if, for any \(A \in \mathcal A\) with \(0 < \mu(A) < \infty\), \(\liminf_N \, {\mu(\bigcup_{n=0}^{N-1} T^{-n} A) \over \mu(\bigcup_{n=0}^{N-1} T^{-n} Y)} \geq 1\). Such a set \(Y\) may exists or not for a C.E.M.P. dynamical system \(\mathsf{T}\). If it exists, the asymptotic equivalence class \(\mathsf{w}(\mathsf{T}) := \{ (w_N)_{N \geq 1} \in (0, \infty): \, w_N \sim w_N(Y) \}\) defines the ``wandering rate'' of \(\mathsf{T}\). For two C.E.M.P. dynamical systems \(\mathsf{T}\) and \(\mathsf{T}^*\), \(\mathsf{T}\) is called a ``\(c\)-factor'' of \(\mathsf{T}^*\) (and \(\mathsf{T}^*\) a ``\(c\)-extension'' of \(\mathsf{T}\)), if there are a measurable map \(\pi\): \(X^* \to X\) and a number \(c \in (0, \infty)\) such that \(\pi \circ T^* = T \circ \pi \text{ and } \mu^* \circ \pi^{-1} = c \, \mu.\) We say that \(\mathsf{t}_1\) and \(\mathsf{t}_2\) are ``similar'', if there are \(c_1, c_2\) in \((0, \infty)\) such that \(\mathsf{t}_1\) and \(\mathsf{t}_2\) have a common extension \(\mathsf{T}^*\) respectively for \(c_1\) and \(c_2\). The main result of the paper is the following theorem: If \(\mathsf{T}\) and \(\mathsf{T}^*\) are C.E.M.P. dynamical systems such that \(\mathsf{T}\) is a \(c\)-factor of \(\mathsf{T}^*\), then \(\mathsf{T}^*\) has a wandering rate if and only if \(\mathsf{T}^*\) has one. In this case, \(\mathsf{w}(\mathsf{T}^*) = c \, \mathsf{w}(\mathsf{T})\). A corollary is that the wandering rate \(\mathsf{w}(\mathsf{T})\) is a similarity invariant.
infinite measure preserving transformation
scientific article; zbMATH DE number 5884602
scientific article