We construct a curve in the unstable foliation of an Anosov diffeomorphism such that the holonomy along this curve is defined on all of the corresponding stable leaves.
For a continuous semicascade on a metrizable compact set Ω, we consider the weak* convergence of generalized operator ergodic means in End C*(Ω). We discuss conditions on the dynamical system under which: every ergodic net contains a convergent sequence; all ergodic nets converge; all ergodic sequences converge. We study the relationships between the convergence of ergodic means and the properties of transitivity of the proximality relation on Ω, minimality of supports of ergodic measures, and uniqueness of minimal sets in the closure of trajectories of a semicascade. These problems are solved in terms of three algebraic-topological objects associated with the dynamical system: the Ellis enveloping semigroup E, the Kohler operator semigroup Г, and the semigroup G that is the weak* closure of the convex hull of Г in End C*(Ω). The main results are stated for semicascades with metrizable E and for tame semicascades.
We consider stationary stochastic processes (Formula presented.) such that (Formula presented.) lies in the closed linear span of (Formula presented.); following Ghosh and Peres, we call such processes linearly rigid. Using a criterion of Kolmogorov, we show that it suffices, for a stationary stochastic process to be linearly rigid, that the spectral density vanishes at zero and belongs to the Zygmund class (Formula presented.). We next give a sufficient condition for stationary determinantal point processes on (Formula presented.) and on (Formula presented.) to be linearly rigid. Finally, we show that the determinantal point process on (Formula presented.) induced by a tensor square of Dyson sine kernels is not linearly rigid.
Thurston parameterized quadratic invariant laminations with a non-invariant lamination, the quotient of which yields a combinatorial model for the Mandelbrot set. As a step toward generalizing this construction to cubic polynomials, we consider slices of the family of cubic invariant laminations defined by a fixed critical leaf with non-periodic endpoints. We parameterize each slice by a lamination just as in the quadratic case, relying on the techniques of smart criticality previously developed by the authors.