A novel theory of quantum physics is developed which synthesises the role of symbolism in two distinct areas of physics: the symbolic algebra of quantum measurement and symbolic dynamics on fractal invariant sets in nonlinear dynamical systems theory. In this synthesis, the universe $U$ is treated as an isolated deterministic dynamical system evolving precisely on a measure-zero fractal invariant subset $I_U$ of its state space. By treating the geometry of $I_U$ as more primitive than differential (or finite-difference) evolution equations on $I_U$, a non-classical approach to the fundamental physics of $U$ is developed. In particular, using symbolic notation, a specific topological representation of $I_U$ is constructed which encodes quaternionic multiplication and from which the statistical properties of the complex Hilbert Space are emergent.
In the realistic setting of Invariant Set Theory, the non-commutativity of Hilbert Space observables is manifest from the number-theoretic incommensurateness of $\phi/\pi$ and $\cos \phi$ for angular coordinate $\phi$; physically this describes the precise sense in which the measure-zero set $I_U$ is counterfactually incomplete. Such incompleteness allows reinterpretations of familiar quantum phenomena, consistent with realism, local causality and effective experimenter free will. By construction, Invariant Set Theory implies the existence of a much stronger synergy between cosmological and quantum physics than is the case in contemporary physical theory. As such, the theory suggests an approach to synthesising gravitational and quantum physics, quite different from current approaches.
As a result, Invariant Set Theory provides new perspectives on key problems (such as the nature of the dark universe and information loss in black holes) in contemporary physics.