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@user539887 not necessarily. In my mental picture all the solutions from minus infinity approach the stable point, For the parameters above, the stable point is close to $x=-10$ and the unstable one is close to $5$. The stable point has a non-compact basin of attraction. If we have diagonal $A$ and coordinate-wise repetition of parameters, the picture persists but now there are some saddle points appearing too.
@user539887 you are right, indeed one has to have some additional assumptions for what I claimed to be true. Some inequalities relating $A, b$ and $\phi$ should be satisfied. In 1D for $A=-1/10$, $b=-1$ and $\phi(x) =\frac{ e^x}{100} $. The conditions in 1D are not difficult to write, but it becomes less obvious in higher dimensions when $A$ is not diagonal.
About the Sarig results -- he proves that the system itself is Bernoulli (with countable alphabet), or that the corresponding induced system is Bernoulli? Infinite alphabet shifts should have infinite entropy, if I understand right and it puzzles me. For the flows there are also Sinai billiards and Lorenz flow, but seemingly Bernoullicity for a flow does not necessarily imply Bernoullicity for its time-1 map (or a Poincare map), although I don't see an immediate example showing why it is so.
About decay of correlations -- yes. But I am asking really about spectral gap. Neither renewal theory nor standard Young approach do not allow to say something about spectral properties of the original system (as far as I understand them). One obtains information about some transfer-like operator (not the one for original system) and then using it obtains some information about decay of correlations for the original system directly. My map is not analytic.
