LaViolette has summarized observational data supporting a non-expanding universe. More recently, Sandage and J-M. Perelmuter have concluded (with reservations) that the universe is expanding based on the analysis of the surface brightness of large elliptical galaxies. These are conflicting studies which do not prove conclusively that the universe is expanding or non-expanding.
To show that the universe is non-expanding, both a viable theory and observational data supporting the theory must exist. Any non-expanding universe theory or model must include, at the very least, a physical process for the Hubble red-shift and a process which prevents the gravitational collapse of the universe. This approach is almost self evident. However, a basic problem appears to exist. There is a long history of futile attempts to explain the Hubble red-shift in a non-expanding universe. If the universe is really not expanding, what do these attempts indicate? I believe these attempts show that current physical concepts are inadequate. Therefore, new physics is needed to develop a non-expanding universe model.