We study Smale skew product endomorphisms (introduced in Mihailescu and Urbański [Skew product Smale endomorphisms over countable shifts of finite type. Ergod. Th. & Dynam. Sys. doi: 10.1017/etds.2019.31. Published online June 2019]) now over countable graph-directed Markov systems, and we prove the exact dimensionality of conditional measures in fibers, and then the global exact dimensionality of the equilibrium measure itself. Our results apply to large classes of systems and have many applications. They apply, for instance, to natural extensions of graph-directed Markov systems. Another application is to skew products over parabolic systems. We also give applications in ergodic number theory, for example to the continued fraction expansion, and the backward fraction expansion. In the end we obtain a general formula for the Hausdorff (and pointwise) dimension of equilibrium measures with respect to the induced maps of natural extensions ${\mathcal{T}}_{\unicode[STIX]{x1D6FD}}$ of $\unicode[STIX]{x1D6FD}$-maps $T_{\unicode[STIX]{x1D6FD}}$, for arbitrary $\unicode[STIX]{x1D6FD}>1$.

We explore the cardinality of generalised inverse limits. Among other things, we show that, for any $n\in \{ℵ_{0},c,1,2,3,\dots \}$, there is an upper semicontinuous function with the inverse limit having exactly $n$ points. We also prove that if $f$ is an upper semicontinuous function whose graph is a continuum, then the cardinality of the corresponding inverse limit is either 1, $ℵ_{0}$ or $c$. This generalises the recent result of I. Banič and J. Kennedy, which claims that the same is true in the case where the graph is an arc.

We investigate inverse limits in the category $ \mathcal{CHU} $ of compact Hausdorff spaces with upper semicontinuous functions. We introduce the notion of weak inverse limits in this category and show that the inverse limits with upper semicontinuous set-valued bonding functions (as they were defined by Ingram and Mahavier [‘Inverse limits of upper semi-continuous set valued functions’, Houston J. Math.  32 (2006), 119–130]) together with the projections are not necessarily inverse limits in $ \mathcal{CHU} $ but they are always weak inverse limits in this category. This is a realisation of our categorical approach to solving a problem stated by Ingram [An Introduction to Inverse Limits with Set-Valued Functions (Springer, New York, 2012)].

We discuss the problem known in economics as backward dynamics occurring in models of perfect foresight, intertemporal equilibrium described mathematically by implicit difference equations. In a previously published paper [Journal of Economic Dynamics and Control 31 (2007), 1633–1671], we showed that by means of certain mathematical methods and results known as inverse limits theory it is possible to establish a correspondence between the backward dynamics of a noninvertible map and the forward dynamics of a related invertible map acting on an appropriately defined space of sequences, each of whose elements corresponds to an intertemporal equilibrium. We also proved the existence of different types of topological attractors for one-dimensional models of overlapping generations. In this paper, we provide an extension of those results, constructing a Lebesgue-like probability measure on spaces of infinite sequences that allows us to distinguish typical from exceptional dynamical behaviors in a measure–theoretical sense, thus proving that all the topological attractors considered in MR07 are also metric attractors. We incidentally also prove that the existence of chaos (in the Devaney–Touhey sense) backward in time implies (and is implied by) chaos forward in time.