The earliest theoretical arguments Shlosman et al. (1989), Pfenniger & Norman (1990) and models (e.g. Friedli & Martinet 1993) suggested that inner bars would be decoupled from - and in fact faster-rotating than - outer bars. Both orbital models (see the contribution by Witold Maciejewski in this volume) and N-body simulations (see the contribution by Juntai Shen in this volume) support this from a theoretical point of view.
From an observational point of view, there is good indirect evidence for decoupled inner bars. The first observational studies of double bars as a class Buta & Crocker (1993), Friedli & Martinet (1993) pointed out that inner and outer bars seemed to be randomly oriented with respect to each other - in particular, inner bars were not found preferentially either perpendicular or parallel to outer bars, which the simplest models of corotating double bars would require. There have been some more sophisticated models of corotating inner bars Shaw et al. (1993), Heller & Shlosman (1996), but even these predict preferred orientations (the inner bar must lead the outer bar).
Figure 6 shows an updated plot of relative position angles between inner and outer bars. As was seen earlier with smaller numbers, inner bars do not preferentially lead or trail outer bars, and the relative angles between them appear to be randomly distributed. This is consistent with the general argument that inner bars rotate independently. Although some recent models Maciejewski & Sparke (2000), Debattista & Shen (2007) suggest that the relative patten speeds of inner bars should vary, so that inner bars spend more time perpendicular to outer bars, this cannot be a very strong effect, as inner bars are not preferentially seen in near-perpendicular orientations.
Figure 6. Deprojected relative positions angles between the bars of double-barred galaxies. Left: position angles in a leading/trailing sense (positive = inner bar leads outer bar), for 52 galaxies where sense of rotation can be determined. Right: absolute position angle between inner and outer bars, for 61 galaxies.
There is limited evidence for decoupled pattern speeds from hydrodynamical modeling of individual galaxies, where one attempts to match the gas flow in potentials with one or more rotating bars to the observed gas kinematics of a particular galaxy. For example, Ann (2001) found a good match for gas morphology when the inner bar of NGC 4314 was rotating faster than the outer bar; however, no kinematic comparison was made. More promising are the cases of NGC 1068 and NGC 4736, though complete, self-consistent modeling for both galaxies is lacking (see the summary in Erwin 2004).
To date, there has been only one published attempt to directly measure pattern speeds in a double-barred galaxy, by Corsini et al. (2003). They applied the Tremaine-Weinberg (T-W) method to NGC 2950 (Figure 1), and were able to show that the two bars did not have the same pattern speed. While a pattern speed for the outer bar was measured, determining a specific speed for the inner bar proved more difficult. The peculiar T-W results for the latter have prompted arguments that the inner bar might actually be counter-rotating (Maciejewski 2006; see also Shen & Debattista 2009). While this is an intriguing possibility, the existing models for forming counter-rotating inner bars Friedli (1996), Davies & Hunter (1997) require the majority of stars in the inner galaxy to counter-rotate, to the point of producing clear reversals in the stellar rotation curve - something not seen in NGC 2950 (compare Figures 6 and 7 of Friedli 1996 with Figure 4 of Corsini et al.). One possible test of the counter-rotating inner-bar hypothesis would be to compare hydrodynamical simulations of prograde and retrograde inner-bar systems with observed gas flows - or even with observed dust lanes - since the resulting gas flows and shocks would presumably be rather different.