We obtain a unique, canonical one-to-one correspondence between the space of marked postcritically finite Newton maps of polynomials and the space of postcritically minimal Newton maps of entire maps that take the form p(z)exp(q(z)) for p(z), q(z) polynomials and exp(z), the complex exponential function. This bijection preserves the dynamics and embedding of Julia sets and is induced by a surgery tool developed by Haïssinsky.

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 EndC*(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.

For a smooth manifold of any dimension greater than one, we present an open set of smooth endomorphisms such that any of them has a transitive attractor with a non-empty interior. These maps are *m*-fold non-branched coverings,*m*≥3. The construction applies to any manifold of the form *S* ^{1×M, where S 1 is the standard circle and Mis an arbitrary manifold.}

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.

We show that, up to topological conjugation,} the equivalence class of a Morse-Smale diffeomorphism without heteroclinic curves on a $3$-manifold is completely defined by an embedding of two-dimensional stable and unstable heteroclinic laminations to a characteristic space.