On Morse conjecture of metric transitivity (Q1177972)

It is known that metric transitivity implies topological transitivity. In 1946 and 1973, M. Morse made a conjecture that the converse theorem was probably true for analytic systems or systems with some degree of analytic regularity. In this paper, the author disproves the Morse's conjecture for differentiable systems and almost everywhere analytic flows in dimension \(\geq 2\). Namely one has the following results: 1) Let \(M\) be a compact manifold of dimension \(\geq 2\) and \(r\geq 1\). If there exists a \(C^ r\) flow with topological transitivity on \(M\), then there exists another \(C^ r\) flow with topological transitivity but without metric transitivity on \(M\). 2) Let \(M\) be a compact analytic manifold of dimension \(\geq 2\). If there exists an analytic flow with topological transitivity on \(M\), then there exists an almost everywhere analytic flow with topological transitivity but without metric transitivity on \(M\). A positive answer to the Morse conjecture is also given in the particular case of an analytic flow on the two-dimensional torus.