Entropy and ergodicity of skew-products over subshifts of finite type and central limit asymptotics
We study various aspects of the dynamics of skew-products (Rand Z-extensions) over a subshift of finite type (ssft). In Chapter I we give the basic definitions and terminology. Conditions are given in Chapter II to ensure ergodicity of the skew-product defined by a function of summable variation with respect to an invariant measure u x ⋋ where u is an ergodic shift-invariant Borel probability measure which is quasi-invariant under finite coordinate changes in the shift space (or under finite block exchanges), and ⋋ denotes Lebesgue measure on R or Z (depending on whether the function is real or integer valued). In Chapter III we define a topological entropy concept for the skew-product (defined by a Holder continuous function f), which is given by the growth rate of periodic orbits of bounded f-weights, and we show that this is the minimum value of the pressure function of f. The asymptotics in the central limit theorem is studied in Chapter IV, for the class of Holder continuous functions defined on a subshift of finite type endowed with a stationary equilibrium state of another HOlder continuous function.