quasiinvariant

A quasiinvariant is a term used in mathematics, particularly in areas like group theory and dynamical systems, to describe a property or object that is "almost" invariant under a certain transformation or group action. It suggests a departure from strict invariance, but one that is small or controlled in some sense.

In the context of group theory, a quasiinvariant measure on a space acted upon by a group is a measure that is preserved up to a multiplicative factor that depends on the group element. This factor is often referred to as the Radon-Nikodym derivative. If the factor were always 1, the measure would be invariant. Quasiinvariant measures are important for studying group actions where strict invariance is not present.

In dynamical systems, a quasiinvariant object might be a curve, a set, or a function that doesn't remain exactly the same under the evolution of the system but undergoes a predictable or bounded transformation. For example, a quasiinvariant measure might describe the long-term distribution of points in a system where phase space volume is not conserved. The concept of quasiinvariance helps to analyze the behavior of systems that are not strictly conservative or exhibit some form of dissipation or growth. The precise definition of "quasiinvariant" can vary depending on the specific mathematical field and the nature of the objects and transformations involved.