Bomfim–Carneiro conjecture on spectral gaps for nonhyperbolic local diffeomorphisms

Let $M$ be a compact Riemannian manifold and let $f:M\rightarrow M$ be a $C^{2}$ local diffeomorphism. A transfer operator associated to a complex continuous potential $\phi:M\rightarrow\mathbb{C}$ is defined by

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

The operator has the spectral gap property on a Banach space $E$ if its spectrum has the form

$$\operatorname{sp}(\mathcal{L})=\{\lambda_1\}\cup\Sigma_1,$$

where $\lambda_1>0$ is a leading eigenvalue with one-dimensional associated eigenspace and there is a number $0<\lambda_0<\lambda_1$ such that $\Sigma_1\subset\{z\in\mathbb{C}:|z|<\lambda_0\}$. Bomfim–Carneiro's conjecture. If $f$ is neither uniformly expanding nor a uniformly hyperbolic diffeomorphism, then there is a suitable potential $\phi$ such that $\mathcal{L}_{f,\phi}$ does not have the spectral gap property on a suitable Banach space, such as a space of Hölder continuous functions, smooth functions, or distributions.

The conjecture proposes that the spectral gap phenomenon underlying standard thermodynamic formalism cannot hold for every suitable potential outside the uniformly expanding or uniformly hyperbolic settings. The supplied text does not state whether the conjecture has been resolved.