The spectral-gap obstruction conjecture for non-uniformly hyperbolic local diffeomorphisms

Let $f:M\rightarrow M$ be a $C^{2}$ local diffeomorphism on a Riemannian manifold $M$. For a potential $\phi$, define the transfer operator by

$$\mathcal{L}_{f,\phi}(g)(x):=\sum_{f(y)=x}e^{\phi(y)}g(y).$$

A suitable Banach space may consist of Hölder continuous or smooth functions. Spectral-gap obstruction conjecture. If $f$ is not a uniformly expanding or uniformly hyperbolic endomorphism, then there exists a suitable potential $\phi$ such that $\mathcal{L}_{f,\phi}$ has no spectral gap property when acting on a suitable Banach space of Hölder continuous or smooth functions.

Uniformly expanding and hyperbolic dynamics with Hölder potentials have a spectral gap and therefore do not exhibit thermodynamic phase transitions. The conjecture proposes that failure of uniform expansion or hyperbolicity can be detected by a suitable potential whose transfer operator lacks a spectral gap, a mechanism connected with the phase transitions studied in the surrounding discussion. Its resolution is not established by the supplied context.