In this paper we consider a smooth dynamical system $f$ and give estimates of the growth rates of vector fields and differential forms in the $L_p$ norm under the action of the dynamical system in terms of entropy, topological pressure and Lyapunov exponents. We prove a formula for the topological entropy $$h_{\rm top}=\lim_{n\to\infty} \frac 1n \log \int \Vert Df_x^n\,^{\wedge}\Vert \,dx,$$ where $Df_x^n\,^{\wedge}$ is a mapping between full exterior algebras of the tangent spaces. An analogous formula is given for the topological pressure.

We consider suspension semi-flows of angle-multiplying maps on the circle for Cr ceiling functions with r≥3. Under a Crgeneric condition on the ceiling function, we show that there exists a Hilbert space (anisotropic Sobolev space) contained in the L2 space such that the Perron–Frobenius operator for the time-t-map acts naturally on it and that the essential spectral radius of that action is bounded by the square root of the inverse of the minimum expansion rate. This leads to a precise description of decay of correlations. Furthermore, the Perron–Frobenius operator for the time-t-map is quasi-compact for a Cr open and dense set of ceiling functions.

We give a simple proof for monotonicity of topological entropy in the quadratic family. We use a spectral property of a Ruelle operator.

We consider a certain analytic function β (t) which is an invariant of finite equivalence between two finite state Markov chains. If two such chains P, Q have the same β-function we wish to prove that they are finitely equivalent. To this end we show that U(t)Pt = QtU (t) has a nontrivial matrix solution U over the ring (exp) of integral combinations of exponential functions. In fact we can force U(t) to be strictly positiveat any specified t0. If U(t) has entries from (exp), the sub-semi-ring of positive integral combinations of exponential functions, then P, Q are finitely equivalent. Many examples reinforce the conjecture that U(t) may always be chosen over (exp) when P, Q have the same β-function. We relate the β-function to topological entropy, measure entropy and information variance.

We introduce the concept of Rokhlin dimension for actions of residually finite groups on $\text{C}^{\ast }$-algebras, extending previous such notions for actions of finite groups and the integers by Hirshberg, Winter and the third author. We are able to extend most of their results to a much larger class of groups: those admitting box spaces of finite asymptotic dimension. This latter condition is a refinement of finite asymptotic dimension and has not previously been considered. In a detailed study we show that finitely generated, virtually nilpotent groups have box spaces with finite asymptotic dimension, providing a large class of examples. We show that actions with finite Rokhlin dimension by groups with finite-dimensional box spaces preserve the property of having finite nuclear dimension when passing to the crossed product $\text{C}^{\ast }$-algebra. We then establish a relation between Rokhlin dimension of residually finite groups acting on compact metric spaces and amenability dimension of the action in the sense of Guentner, Willett and Yu. We show that for free actions of infinite, finitely generated, nilpotent groups on finite-dimensional spaces, both these dimensional values are finite. In particular, the associated transformation group$\text{C}^{\ast }$-algebras have finite nuclear dimension. This extends an analogous result about $\mathbb{Z}^{m}$-actions by the first author to a significantly larger class of groups, showing that a large class of crossed products by actions of such groups fall under the remit of the Elliott classification programme. We also provide results concerning the genericity of finite Rokhlin dimension, and permanence properties with respect to the absorption of a strongly self-absorbing $\text{C}^{\ast }$-algebra.

There exists a Bernoulli shift with non-identical factor measures for which no invariant σ-finite equivalent measure exists.

It is proved that all special flows over a rotation by an irrational $\alpha$ with bounded partial quotients and under f which is piecewise absolutely continuous with a non-zero sum of jumps are mildly mixing. Such flows are also shown to enjoy a condition that emulates the Ratner condition introduced in M. Ratner (Horocycle flows, joinings and rigidity of products. Ann. of Math.118 (1983), 277–313). As a consequence we construct a smooth vector-field on $\mathbb{T}^2$ with one singularity point for which the corresponding flow $(\varphi_t)_{t\in\mathbb{R}}$ preserves a smooth measure, its set of ergodic components consists of a family of periodic orbits and one component of positive measure on which $(\varphi_t)_{t\in\mathbb{R}}$ is mildly mixing and is spectrally disjoint from all mixing flows.

We consider families of differentiable diffeomorphisms with hyperbolic points, close to the identity, which tend to it when the parameter goes to zero.

We study the asymptotic behaviour of the invariant manifolds. Then we consider the case when there are homo-heteroclinic points and we find that the maximum separation between the invariant manifolds is of the order of some power of the parameter which is related to the degree of differentiability.

Finally the analogous case for flows is considered.

We examine generalizations of R. Mañé's results on the topological dimension of spaces supporting an expansive homeomorphism to the case of real-expansive flows. We show that a space supporting a real-expansive flow must be finite dimensional, and a minimal real-expansive flow not exhibiting a type of spiral behaviour must be one-dimensional. This latter class includes all known examples and a slight generalization of Axiom A flows. These results are obtained by introducing a new concept of stable and unstable sets for real flows, and examining real-expansive flows in terms of these sets.

We consider unimodular random rooted trees (URTs) and invariant forests in Cayley graphs. We show that URTs of bounded degree are the same as the law of the component of the root in an invariant percolation on a regular tree. We use this to give a new proof that URTs are sofic, a result of Elek. We show that ends of invariant forests in the hyperbolic plane converge to ideal boundary points. We also note that uniform integrability of the degree distribution of a family of finite graphs implies tightness of that family for local convergence, also known as random weak convergence.

For a closed lamination on the unit circle invariant under z\mapsto z^d we prove an inequality relating the number of points in the ‘gaps’ with infinite pairwise disjoint orbits to the degree; in particular, this gives estimates on the cardinality of any such ‘gap’ as well as on the number of distinct grand orbits of such ‘gaps’. As a tool, we introduce and study a dynamically defined growing tree in the quotient space. We also use our techniques to obtain for laminations an analog of Sullivan's no wandering domain theorem. Then we apply these results to Julia sets of polynomials.

We prove that for all rational functions f on the Riemann sphere and potential $-t\ln|f'|, t\ge 0$ all the notions of pressure introduced in Przytycki (Proc. Amer. Math. Soc.351(5) (1999), 2081–2099) coincide. In particular, we get a new simple proof of the equality between the hyperbolic Hausdorff dimension and the minimal exponent of conformal measure on a Julia set. We prove that these pressures are equal to the pressure defined with the use of periodic orbits under an assumption that there are not many periodic orbits with Lyapunov exponent close to 1 moving close together, in particular under the Topological Collet–Eckmann condition. In Appendix A, we discuss the case t < 0.

We introduce and study a non-uniform hyperbolicity condition for complex rational maps that does not involve a growth condition. We call this condition backward contraction. We show this condition is weaker than the Collet–Eckmann condition, and than the summability condition with exponent one. Our main result is a connecting lemma for backward-contracting rational maps, roughly saying that we can perturb a rational map to connect each critical orbit in the Julia set with an orbit that does not accumulate on critical points. The proof of this result is based on Thurston's algorithm and some rigidity properties of quasi-conformal maps. We also prove that the Lebesgue measure of the Julia set of a backward-contracting rational map is zero, when it is not the whole Riemann sphere. The basic tool of this article is sets having a Markov property for backward iterates that are holomorphic analogues of nice intervals in real one-dimensional dynamics.

We prove a formula for the sofic entropy of expansive principal algebraic actions of residually finite groups, extending recent work of Deninger and Schmidt.

For hyperbolic systems and for Hölder cocycles with values in a compact metric group, we extend Livšic's periodic point characterisation of coboundaries. Here we show that two such cocycles are cohomologous when their respective ‘weights’ (of closed orbits) coincide. When it is only assumed that they are conjugate, one of the cocycles must (in general) be modified by an isomorphism (which stabilises conjugacy classes) to obtain cohomology. When the group is Lie and when a transitivity condition is satisfied, conjugacy of weights ensures that the cocycles are cohomologous with respect to a finitely extended group.

We prove the L2-convergence for the linear multiple ergodic averages of commuting transformations $T_{1}, \dots, T_{l}$, assuming that each map Ti and each pair TiTj-1 is ergodic for $i\neq j$. The limiting behavior of such averages is controlled by a particular factor, which is an inverse limit of nilsystems. As a corollary we show that the limiting behavior of linear multiple ergodic averages is the same for commuting transformations.

In this paper we show that a continuous map f from a connected graph G to itself is pointwise-recurrent if and only if one of the following two statements holds: (1) G is a circle, and f is a homeomorphism topologically conjugate to an irrational rotation of the unit circle S1; (2) f is a periodic homeomorphism.

In 1995, Hill and Velani introduced the ‘shrinking targets’ theory. Given a dynamical system ([0,1],T), they investigated the Hausdorff dimension of sets of points whose orbits are close to some fixed point. In this paper, we study the sets of points well approximated by orbits {Tnx}n≥0, where Tis an expanding Markov map with a finite partition supported by [0,1]. The dimensions of these sets are described using the multifractal properties of invariant Gibbs measures.

For a smooth ℤ2-action on a C∞ compact Riemannian manifold M, we discuss its ergodic properties which include the decomposition of the tangent space of M into subspaces related to Lyapunov exponents, the existence of Lyapunov charts, and the subadditivity of entropies.

For any $C^1$ diffeomorphism with dominated splitting, we consider a non-empty set of invariant measures that describes the asymptotic statistics of Lebesgue-almost all orbits. They are the limits of convergent subsequences of averages of the Dirac delta measures supported on those orbits. We prove that the metric entropy of each of these measures is bounded from below by the sum of the Lyapunov exponents on the dominating sub-bundle. As a consequence, if those exponents are non-negative, and if the exponents on the dominated sub-bundle are non-positive, those measures satisfy the Pesin entropy formula.