Universal Non‐Invertible Symmetries

Torus breakdown in noninvertible maps.

We propose a criterion for the destruction of a two-dimensional torus through the formation of an infinite set of cusp points on the closed invariant curves defining the resonance torus. This mechanism is specific to noninvertible maps. The cusp points arise when the tangent to the torus at the point of intersection with the critical curve L(0) coincides with the eigendirection corresponding to...

Homoclinic tangles for noninvertible maps

Not all naturally occurring iterated systems have the property that time is reversible. In which case, the dynamics must be modelled by a noninvertible map rather than by a di eomorphism. While one-dimensional theory describes dynamics of noninvertible maps, higher-dimensional theory has focussed on di eomorphisms, though there are many higher-dimensional examples of natural systems modelled by...

Why Are Symmetries a Universal Language of Physics? (on the unreasonable effectiveness of symmetries in physics)

In this paper, we prove that in some reasonable sense, every possible physical law can be reformulated in terms of symmetries. This result explains the wellknown success of group-theoretic approach in physics. 1 Formulation of the Problem Traditional physics used differential equations to describe physical laws. Modern physical theories (starting from quarks) are often formulated not in terms o...

Entropy for Noninvertible Transformations

Universal local symmetries and nonsuperposition in classical mechanics.

In the Hilbert space formulation of classical mechanics, pioneered by Koopman and von Neumann, there are potentially more observables than in the standard approach to classical mechanics. In this Letter, we show that actually many of those extra observables are not invariant under a set of universal local symmetries which appear once the Koopman and von Neumann formulation is extended to includ...