  We introduce self-similar versions of the one-dimensional almost Mathieu
operators. Our definition is based on a class of self-similar Laplacians
instead of the standard discrete Laplacian, and includes the classical almost
Mathieu operators as a particular case. Our main result establishes that the
spectra of these self-similar almost Mathieu operators can be completely
described by the spectra of the corresponding self-similar Laplacians through
the spectral decimation framework used in the context of spectral analysis on
fractals. In addition, the self-similar structure of our model provides a
natural finite graph approximation model. This approximation is not only
helpful in executing the numerical simulation, but is also useful in finding
the spectral decimation function via Schur complement computations of given
finite-dimensional matrices. The self-similar Laplacians used in our model were
considered recently by Chen and Teplyaev who proved the emergence of singularly
continuous spectra for specific parameters. We use this result to arrive at
similar conclusions in the context of the self-similar almost Mathieu
operators. Finally, we derive an explicit formula of the integrated density of
states of the self-similar almost Mathieu operators as the weighted pre-images
of the balanced invariant measure on a specific Julia set.
