Let $X$ be a compact metric space and let $f\colon X\to X$ define some dynamics, $f$ being continuous and finite-to-one on a global attractor $A\subset X$. Moreover, $A$ is the non-wandering set $\Omega(f)$ of $f$ and the dynamics on $A$ are chaotic (due to the definition of chaos by Devaney). Let $h(f)$, the topological entropy of $f$, be finite.
I am searching for some theorems, contexts, theories etc. that tell me something about the existence of some measures of maximal entropy (or even an unique measure of maximal entropy) in this situation (in particular, the fact that on A, $f$ is finite-to-one seems to be important; I cannot say exactly why).
Maybe you just have some ideas or knowledge that might be helpful or can show me some possible directions I could go. Maybe you can recommend some improvements for my question or books etc.
Thanks in advance!
You really need to be more detailed on what is your setup. For example, any $C^\infty$ map of a compact manifold has measures of maximal entropy, as a consequence of the upper-semicontinuity of the entropy map. In some situations less smoothness is sufficient in full generality, for the existence of measures of maximal entropy, such as for interval maps when the entropy is sufficiently large (in which case the number of measures of maximal entropy is in fact finite). In a somewhat different direction any expansive map of a compact metric space has also an upper-semicontinuous entropy. On the uniqueness, even some topologically mixing subshifts (not topological Markov chains nor finite factors) may have more than one measure of maximal entropy.