Orbits of turning points for maps of finite graphs and inverse limit spaces (Q2782225)

The topology of inverse limit spaces generated by maps of finite graphs is examined. The definition of a Markov map of the interval is extended to a Markov map of a graph and sufficient conditions are found under which the inverse limits of two Markov maps of a graph are homeomorphic. Here the structure of the orbits of the turning points plays the key role. As a corollary, an extension of a theorem of \textit{S. E. Holte} [Pac. J. Math. 156, No. 2, 297-306 (1992; Zbl 0723.58034)] for interval Markov maps is obtained. Then, under some assumptions on a map of a graph it is shown that many of the points \((x_0,x_1, \dots)\) of the corresponding inverse limit have neighbourhoods homeomorphic to the product of a zero-dimensional set and the open interval \((0,1)\). Each of these points has the property that for some positive integer \(m\), the point \(x_m\) does not belong to the \(\omega\)-limit set of the turning points. Again, a corollary for interval maps is obtained.NEWLINENEWLINEFor the entire collection see [Zbl 0983.00046].