  Dynamical systems are abstract models of interaction between space and time.
They are often used in fields such as physics and engineering to understand
complex processes, but due to their general nature, they have found
applications for studying computational processes, interaction in multi-agent
systems, machine learning algorithms and other computer science related
phenomena. In the vast majority of applications, a dynamical system consists of
the action of a continuous 'transition function' on a metric space. In this
work, we consider decidable formal systems for reasoning about such structures.
Spatial logics can be traced back to the 1940's, but our work follows a more
dynamic turn that these logics have taken due to two recent developments: the
study of the topological mu-calculus, and the the integration of linear
temporal logic with logics based on the Cantor derivative. In this paper, we
combine dynamic topological logics based on the Cantor derivative and the 'next
point in time' operators with an expressively complete fixed point operator to
produce a combination of the topological mu-calculus with linear temporal
logic. We show that the resulting logics are decidable and have a natural
axiomatisation. Moreover, we prove that these logics are complete for
interpretations on the Cantor space, the rational numbers, and subspaces
thereof.
