Analytic Paths of Ergodic Measures with Prescribed Invariants

We study analytic paths of ergodic measures under quantitative constraints. For uniformly hyperbolic systems, we construct one-parameter families of ergodic measures whose prescribed Birkhoff averages vary affinely, whose metric entropies vary analytically, and whose endpoint entropy values are realized exactly. Along these paths, the integral of every Hölder continuous observable depends analytically on the parameter. In the mixing case, the measures may be chosen to be Bernoulli. We also prove a dimension counterpart for average conformal hyperbolic sets: the Hausdorff dimensions of the measures vary analytically along the path. Finally, for a class of partially hyperbolic diffeomorphisms with one-dimensional center, we construct analytic paths of ergodic measures whose center Lyapunov exponents are prescribed linearly and whose entropies vary nalytically.