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5. Other Directions


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      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 leaf-wise conditional measures \mu^{\mathcal W}_\varphi which satisfy Margulis-like cocycle properties?


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          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.


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              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?


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                  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, Weil-Petersson geodesic flows, noncompact spaces, non-positive curvature, etc)


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                      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?


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                          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.


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                              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?


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                                  Problem 5.8.

                                  [Mark Demers] Consider a finite-horizon 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?


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                                      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.