5. Other Directions

Problem 5.1.
[Kurt Vinhage] Let $f_1,f_2$ be commuting hyperbolic diffeomorphisms of a compact manifold $M$. One may construct coarse Lyapunov foliations $\mathcal W$. Given a potential $\varphi$, can one construct leafwise conditional measures $\mu^{\mathcal W}_\varphi$ which satisfy Margulislike cocycle properties? 
Problem 5.2.
[Dan Thompson] For metric Anosov flows (Smale flows), under what conditions does orbit equivalence imply Hölder orbit equivalence? Can one find a Smale flow which is not Hölder covered by the suspension of a symbolic dynamical system. 
Problem 5.3.
[Kurt Vinhage] Let $f : \mathbb{T}^2 \to \mathbb{T}^2$ be a nonuniformly hyperbolic diffeomorphism. Let $\mu$ be the unstable SRB measure, $\nu$ be the measure of maximal entropy and $m$ be the measure of maximal dimension (make assumptions on $f$ so that these exist and are unique). If $\nu = \mu$ or $m$, is $f$ smoothly conjugate to a linear automorphism? 
Problem 5.4.
[Jayadev Athreya] Can one prove exponential decay of correlations for equilibrium states in settings where it it is known for the measure of maximal entropy? (eg, WeilPetersson geodesic flows, noncompact spaces, nonpositive curvature, etc) 
Problem 5.5.
[Jerome Buzzi] Can one find a dynamical interpretation of the pressure gap? For instance, can one relate the pressure gap to an essential spectral radius for a dynamical operator? Or some dynamically defined $\zeta$function? 
Problem 5.6.
[Kiho Park] Describe the set of matrix cocycles which have multiple eqilibrium states. For simplicity, consider a subshift of finite type, and a locally constant, fiber bunched cocycle. 
Problem 5.7.
[Vaughn Climenhaga] build a moduli space for $C^{1,\theta}$perturbations of geodesic flows using potential functions (following McMullen’s approach of expanding maps of the circle). In this case, can one identify the space of geodesic flows by a condition on their corresponding potential function? 
Problem 5.8.
[Mark Demers] Consider a finitehorizon billiards, and let $\varphi$ be the geometric potential. Is there a jump in pressure $P(q\varphi)$ and $q \to 0^$? Can one find perioidic orbits with arbitrarily large (but finite) Lyapunov exponent? 
Problem 5.9.
[Kiho Park] Develop the theory for weighting orbits in a subadditive (rather than additive) way. {m ̊Ask Kiho for references}
Cite this as: AimPL: Equilibrium states for dynamical systems arising from geometry, available at http://aimpl.org/equibdynsysgeom.