A common observation in data-driven applications is that data has a low intrinsic dimension, at least locally. Thus, when one wishes to work with data that is not governed by a clear set of equations, but still wishes to perform statistical or other scientific analysis, an optional model is the assumption of an underlying manifold from which the data was sampled. This manifold, however, is not given explicitly but we can obtain samples of it (i.e., the individual data points).
Rigidity of horospherical group actions, and more generally unipotent group actions, is a well established phenomenon in homogeneous dynamics. Whereas all finite ergodic horospherically invariant measures are algebraic (due to Furstenberg, Dani and Ratner), the category of locally finite measures, particularly in the context of geometrically infinite quotients, is known to be much richer (following works by Babillot, Ledrappier and Sarig). The rigidity of such locally finite measures is manifested in them having large and exhaustive stabilizer groups.
Motivated by questions in p-adic Fourier theory, we study invariant norms on the p-adic Schrödinger representations of Heisenberg groups. These Heisenberg groups are p-adic, and the Schrödinger representations are explicit irreducible smooth representations that play an important role in their representation theory.
A power series f is said to satisfy a p-Mahler equation (p>1 a natural number) if it satisfies a functional equation of the form a_n(x).f(x^{p^n}) + ... + a_1(x).f(x^p) + a_0(x).f(x) = 0 where the coefficients a_i(x) are polynomials. These functional equations were studied by Kurt Mahler with relation to transcendence theory.
