We study a class of open chaotic dynamical systems. Consider an expanding map of an interval from which a few small open subintervals are removed (thus creating ‘holes’). Almost every point of the original interval then eventually escapes through the holes, so there can be no absolutely continuous invariant measures. We construct a so-called conditionally invariant measure that is equivalent to the Lebesgue measure. Our measure is unique and naturally generates an invariant measure, which is singular. These results generalize early work by Pianigiani, Yorke, Collet, Martinez, and Schmidt, who studied similar maps under an additional Markov assumption. We do not assume any Markov property here and use ‘bounded variation’ techniques rather than Markov coding. Our results supplement those of Keller, who studied analytic interval maps with holes by using different techniques.

Let f be a rational function with a fixed point z_0 of multiplicity m+1. Then there are m invariant components of the Fatou set, called immediate parabolic basins, where the iterates of f tend to z_0, and each immediate parabolic basin contains at least one critical point of f. A function \phi holomorphic and univalent near z_0 with \phi(z_0)=0 such that h:=\phi\circ f\circ\phi^{-1} satisfies h(z)=z-z^{m+1}+\alpha z^{2m+1}+O(z^{2m+2}) as z\to 0 for some \alpha\in\mathbb{C} also exists. This number \alpha is invariant under holomorphic changes of coordinates. We show that if each immediate parabolic basin at z_0 contains exactly one critical point of f, and if this critical point is simple, then \textrm{Re}\alpha\leq \frac14m+\frac12. We also discuss the case that the critical points in the immediate parabolic basins are multiple and establish an upper bound for the real part of \alpha depending on m and the multiplicities of the critical points contained in the parabolic basins.

We show that in a topological dynamical system (X,T) of positive entropy there exist proper (positively) asymptotic pairs, that is, pairs (x,y) such that x\not= y and \lim_{n\to +\infty} d(T^n x,T^n y)=0. More precisely we consider a T-ergodic measure \mu of positive entropy and prove that the set of points that belong to a proper asymptotic pair is of measure one. When T is invertible, the stable classes (i.e. the equivalence classes for the asymptotic equivalence) are not stable under T^{-1}: for \mu-almost every x there are uncountably many y that are asymptotic to x and such that (x,y) is a Li–Yorke pair with respect to T^{-1}. We also show that asymptotic pairs are dense in the set of topological entropy pairs.

For diffeomorphisms on 3-manifolds, two-dimensional links composed of tori (and possibly one Klein bottle) embedded in S^2\times S^1 appear in a natural way to describe the topological position of two-dimensional unstable manifolds of saddles in the basin of a periodic sink. For such a link \mathcal{E} we build a diffeomorphism f_{\mathcal{E}} which is used as a canonical model for ‘pieces of the dynamics’, namely the basin of attractors consisting of a sink and a finite set of two-dimensional unstable manifolds of saddles.

Let G be a discrete group which admits an amenable action on a compact space and \gamma \in \textrm{Aut}(G) be an automorphism. We define a notion of entropy for \gamma and denote the invariant by ha(\gamma). This notion is dual to classical topological entropy in the sense that if G is abelian then ha(\gamma) = h_{\rm Top}(\hat{\gamma}) where h_{\rm Top}(\hat{\gamma}) denotes the topological entropy of the induced automorphism \hat{\gamma} of the (compact, abelian) dual group \hat{G}.

ha(\cdot) enjoys a number of basic properties which are useful for calculations. For example, it decreases in invariant subgroups and certain quotients. These basic properties are used to compute the dual entropy of an arbitrary automorphism of a crystallographic group.

In singular perturbation problems one frequently has a family (\pi_\eps)_{\eps\in]0,1]} of semiflows on a space X and a ‘singular limit flow’ \pi_0 of this family which is defined only on a subspace X_0 of X. Moreover, \pi_\epsilon converges to \pi_0 only in some ‘singular’ sense. Such a situation occurs, for example, in fast–slow systems of differential equations or in evolution equations on thin spatial domains.

In this paper we prove a general singular Conley index continuation principle stating that every isolated invariant set K_0 of \pi_0 can be continued to a nearby family K_\epsilon of isolated invariant sets of \pi_\epsilon with the same Conley index. We illustrate this continuation result with damped wave equations on thin domains. This extends some results from our previous work.

Relative transfer operators are used in Denker and Gordin's 1999 paper to construct Gibbs families of conditional measures on fibres of a uniformly-expanding and exact-fibred system. Here we investigate the relationship between these operators and the relative variational principle (extending results for non-fibred expanding systems).

It turns out that the maximal value for the free energy in the relative variational problem can be represented in terms of the transfer operator. However, for a general potential, the possibility to reduce the construction of an equilibrium measure to the search for an appropriate family of conditional measures on the fibres, critically depends on the invertibility of the base transformation.

A certain class of potentials (called basic) which allow the last-mentioned reduction is introduced and the properties of the corresponding equilibrium measures are studied. Any measure of this kind gives rise to a regular factor; under a natural assumption the latter property is shown to be equivalent to the validity of the relative version of Rokhlin's formula for the entropy of a measure preserving transformation. Several examples are presented, among them families of polynomial skew products in \mathbb{C}^n restricted to their Julia sets.

We refine the transformation to smooth normal form for an analytic family of vector fields in the neighbourhood of a saddle-node. This refinement is very powerful and allows us to prove the finite cyclicity of families of graphics (‘ensembles’) occurring inside analytic families of vector fields. In [ZR1] and [ZR2] it is used to prove the finite cyclicity of graphics through a nilpotent singular point of elliptic type. Several examples are presented: lips, graphics with two subsequent lips, graphics with a nilpotent point of elliptic type and a saddle-node. We also discuss the bifurcation diagram of limit cycles for a graphic in the lips.

We prove the existence of periodic orbits for steady real-analytic Euler flows on all Riemannian solid tori. By using the correspondence theorem from part I of this series, we reduce the problem to the Weinstein Conjecture for solid tori. We prove the Weinstein Conjecture on the solid torus via a combination of results due to Hofer et al and a careful analysis of tight contact structures on solid tori.

In this paper we consider some algebraic structure of groups of real-analytic diffeomorphisms of the circle.

We present a certain version of the ‘non-stationary’ normal forms theory for extensions of topological dynamical systems by smooth contractions of \mathbb{R}^n. The central concept is a notion of a sub-resonance relation which is an appropriate generalization of the notion of resonance between eigenvalues. Finitely differentiable, infinitely differentiable and analytic maps are considered.

We compute the exact lower bounds for some averages arising in ergodic theory. In particular, we prove that for any measure-preserving system (X,\mathcal{B},\mu,T) with \mu(X)<\infty, any A\in\mathcal{B} and any N\in\mathbb{N}, N^{-1}\sum_{n=0}^{N-1}\mu(A\cap T^{-n}A)\geq\sqrt{\mu(A)^{2}+(\mu(X)-\mu(A))^{2}}+\mu(A)-\mu(X).

Any Axiom A holomorphic map on \mathbb{P}^2\mathbb{C} has a natural ergodic measure \mu. The set K^- is the set of points which have arbitrarily close neighbors with prehistories not convergent to supp \mu.

This set is the analogue of the set of points with bounded backwards iterates from the case of Hénon mappings. For Hénon diffeomorphisms it was shown by Bedford and Smillie that K^- either has an empty interior or \mathop{K}\limits^\circ{}^{-}= union of the basins of finitely many repelling periodic points. For s-hyperbolic holomorphic endomorphisms of \mathbb{P}^2, we show here that the only possibility is \mathop{K}\limits^\circ{}^{-} = \emptyset. This answers a question of Fornaess.

We also prove that \mathop{K}\limits^\circ{}^{-} is the union of finitely many repelling basins when the endomorphism is a perturbation of a Hénon mapping.

Several non-trivial examples of s-hyperbolic maps are discussed as well, some of them coming from perturbations of product maps and others from solenoids.

An extension of the Shannon–McMillan–Breiman Theorem to a class of non-commutative dynamical systems is given.

The linear action of SL (n, \mathbb{Z}) on \mathbb{R}^n has non-trivial invariant sets and matrices of SL (n, \mathbb{Z}) preserve volume. Here we prove that a Borel measure on \mathbb{R}^n, locally finite at a point, invariant under the linear action of SL (n, \mathbb{Z}) annihilating the set \{tz: t\geq 0,z\in \mathbb{Z}^n\} must be a scalar multiple of the Lebesgue measure. Our result extends a theorem due to S. G. Dani concerning locally finite measures, ergodic under the linear action of SL (n, \mathbb{Z}). Our approach is alternative, it exploits the Euclidean structure of the space and the fact that the probability of n integers being relatively prime is strictly positive.

In this paper we consider symplectic and volume-preserving diffeomorphisms of compact manifolds, and prove that, C^{r} generically, 1\leq r\leq\infty, the invariant manifolds of hyperbolic periodic points contain a dense subset of recurrent points. As a consequence, each of these invariant manifolds is contained in its omega limit set, and its closure is a chain transitive set.

We use methods from conformal geometry to give an explicit estimate for how many iterations of Newton's method it takes, at most, to find all the roots of an arbitrary complex polynomial of fixed degree with prescribed precision.

We exhibit an example of a continuous map of the interval of type 2^\infty, and hence of zero topological entropy, for which the set of periodic points is not a G_\delta set. This disproves a conjecture by Sharkovsky from 1965. Unfortunately, this conjecture has been incorrectly quoted as a true statement by other authors in many papers and books.