The purpose of this paper is to survey the key ideas which have played a role in the development of the theory of smooth unimodal maps in the 1990s so as to provide a starting point for more detailed study. Most papers underlying this survey are distinguished by great technical complexity; we have therefore attempted to extract their essence and present it in a way accessible to a non-specialist. For further study we compiled a long list of original references. Another goal of this paper is to present the multi-directional process in which ideas flowed and developed, placed in a historical order.

We consider the set of finite Blaschke products $F$ for which the fixed points on the circle $S^1$ are expanding and we prove that if $F'(x) \ne F'(y)$ for all different fixed points $x,y$ of $F$ on $S^1$, then $F$ commutes only with its own powers.

Consider a $C^{1+\epsilon}$ diffeomorphism $f$ having a uniformly hyperbolic compact invariant set $\Omega$, maximal invariant in some small neighbourhood of itself. The asymptotic exponential rate of escape from any small enough neighbourhood of $\Omega$ is given by the topological pressure of $-\log |J^u f|$ on $\Omega$ (Bowen–Ruelle in 1975). It has been conjectured (Eckmann–Ruelle in 1985) that this property, formulated in terms of escape from the support $\Omega$ of a (generalized Sinai–Ruelle–Bowen (SRB)) measure, using its entropy and positive Lyapunov exponents, holds more generally. We present a simple $C^\infty$ two-dimensional counterexample, constructed by a surgery using a Bowen-type hyperbolic saddle attractor as the basic plug.

It is a known fact that an iterated function system (IFS) of entire functions is not necessarily ergodic. In this paper we show that if an IFS of analytic functions is defined in a domain whose boundary contains more than two points (in the extended complex plane) then the system possesses an ergodic property.

We define staircase $\mathbb{Z}^d$ actions. We first prove that staircase $\mathbb{Z}^2$ actions satisfying a general condition are mixing. Then we describe how to extend the results to the staircase $\mathbb{Z}^d$ actions. Thus we have constructed explicitly rank one mixing $\mathbb{Z}^d$ actions which include natural analogues to the well-known staircase transformation.

We show that the Poincaré series of the Fuchsian group of deck transformations of ${\mathbb C}\setminus{\mathbb Z}$ diverges logarithmically. This is because ${\mathbb C}\setminus{\mathbb Z}$ is a ${\mathbb Z}$-cover of the three horned sphere, whence its geodesic flow has a good section which behaves like a random walk on ${\mathbb R}$ with Cauchy distributed jump distribution and has logarithmic asymptotic type.

Suppose that $F$ is a $C^\infty$ diffeomorphism of the plane with hyperbolic fixed point $p$ for which a branch of the unstable manifold, $W^u_+(p)$, has a same-sided quadratic tangency with the stable manifold, $W^s(p)$. If the eigenvalues of $DF$ at $p$ satisfy a non-resonance condition, each nonempty open set of $ \cl( W^u_+(p))$ contains a copy of any continuum that can be written as the inverse limit space of a sequence of unimodal bonding maps.

In this paper we prove, using explicit constructions, that every non-compact manifold (except, of course, surfaces of finite genus) can be endowed with a non-singular flow without minimal subset, that is to say: a flow such that each orbit closure contains a smaller one.

We prove that the set of linear cocycles with simple Lyapunov spectrum is dense in the space of all linear cocycles equipped with the uniform topology.

Let $f$ be a holomorphic self-map of $\mathbb{C} \backslash \{ 0 \}, \mathbb{C}$, or the extended complex plane $\overline{\mathbb{C}}$ that is neither injective nor constant. This paper gives new and elementary proofs of the well-known fact that the Julia set of $f$ is a non-empty perfect set and coincides with the closure of the set of repelling cycles of $f$. The proofs use Montel–Caratheodory's theorem but do not use results from Nevanlinna theory.

Let $f: \mathbb{R}^{2} \rightarrow \mathbb{R}^{2}$ be a homeomorphism of the plane. We define open sets $P \subset \mathbb{R}^{2}$, called pruning fronts after the work of Cvitanović et al. in 1988, for which it is possible to construct an isotopy $H: \mathbb{R}^{2} \times [0,1] \rightarrow \mathbb{R}^{2}$ with open support contained in $\bigcup _{n \in {\mathbb{Z}} } f^{n} (P)$ such that $H(\cdot, 0 ) = f(\cdot)$ and $H(\cdot, 1) = f_{P} (\cdot)$, where $f_P$ is a homeomorphism under which every point of $P$ is wandering. Applying this construction when $f$ is Smale's horseshoe, it is possible to obtain an uncountable family of homeomorphisms, depending on infinitely many parameters, going from trivial to chaotic dynamic behavior. This family is a two-dimensional analog of a one-dimensional universal family.

Chaotic and ergodic properties are discussed for various subclasses of cylindric billiards. A common feature of the studied systems is that they satisfy a natural necessary condition for ergodicity and hyperbolicity, the so-called transitivity condition. The relation of our discussion to former results on hard ball systems is twofold. On the one hand, by slight adaptation of the proofs we may discuss hyperbolic and ergodic properties of 3 or 4 particles with (possibly restricted) hard ball interactions in any dimensions. On the other hand, a key tool in our investigations is a kind of connected path formula for cylindric billiards, which together with the conservation of momenta gives back, when applied to the special case of hard ball systems, the classical connected path formula.

Given a Lie group $G$ and a lattice $\Gamma$ in $G$, a one-parameter subgroup $\phi$ of $G$ is said to be rigid if for any other one-parameter subgroup $\psi$, the flows induced by $\phi$ and $\psi$ on $\Gamma\backslash G$ (by right translations) are topologically orbit-equivalent only if they are affinely orbit-equivalent. It was previously known that if $G$ is a simply connected solvable Lie group such that all the eigenvalues of $\mathrm{Ad} (g) $, $g\in G$, are real, then all one-parameter subgroups of $G$ are rigid for any lattice in $G$. Here we consider a complementary case, in which the eigenvalues of $\mathrm{Ad} (g)$, $g\in G$, form the unit circle of complex numbers.

Let $G$ be the semidirect product $N \rtimes M$, where $M$ and $N$ are finite-dimensional real vector spaces and where the action of $M$ on the normal subgroup $N$ is such that the center of $G$ is a lattice in $M$. We prove that there is a generic class of abelian lattices $\Gamma$ in $G$ such that any semisimple one-parameter subgroup $\phi$ (namely $\phi$ such that $\mathrm{Ad} (\phi_t)$ is diagonalizable over the complex numbers for all $t$) is rigid for $\Gamma$ (see Theorem 1.4). We also show that, on the other hand, there are fairly high-dimensional spaces of abelian lattices for which some semisimple $\phi$ are not rigid (see Corollary 4.3); further, there are non-rigid semisimple $\phi$ for which the induced flow is ergodic.

An example is given of a one-dimensional foliation (or a flow) on an open surface which admits no minimal set.

In a recent article, Manouchehri proved a ‘Sternberg theorem’ for Liouville vector fields and noticed that it provided normal forms for implicit differential equations and first-order partial differential equations. We establish local and global versions of Moser's celebrated result on volume and symplectic forms when they admit a non-trivial one parameter (pseudo-) group of homotheties—by definition, a Liouville field is the generator of such a flow. The local version implies that two germs of Liouville fields of a symplectic or volume form are conjugate by a diffeomorphism germ which preserves the form if and only if they are conjugate. This contains Manouchehri's theorem.

For parameter values $a$ where the quadratic map $f_a(x) = 1-ax^2$ has an attracting periodic point, and small values of $b$, the Hénon map $H_{a,b}(x, y) = (1 + y - ax^2, bx)$ has a periodic attractor and an attracting set that is homeomorphic with the inverse limit of $f_a|_{[1-a, 1]}$. This attracting set consists of the collection of unstable manifolds of a hyperbolic invariant set together with the attracting periodic orbit. In the case in which $f_a$ is also not renormalizable with smaller period and $b < 0$, we give a symbolic description of the collection of stable manifolds of this hyperbolic set and show that this collection is homeomorphic with the collection of unstable manifolds precisely when the attracting periodic orbit is accessible from the complement of the attracting set, a condition that can be characterized in terms of the kneading sequence of the quadratic map. As an application, we answer a question raised by Hubbard and Oberste-Vorth by proving that the basin boundaries corresponding to three distinct period five sinks in the Hénon family are non-homeomorphic.

We consider in this paper the space of all orientation preserving and fixed point free homeomorphisms of the plane, endowed with the compact-open topology. Our main theorem asserts that this space is locally contractible. Then we give a new proof of the path connectedness. Finally, it is shown that the inclusion of this space into the space of all orientation preserving planar homeomorphisms induces the trivial map on the fundamental groups.

This paper gives two results that show that the dynamics of a time-periodic Lagrangian system on a hyperbolic manifold are at least as complicated as the geodesic flow of a hyperbolic metric. Given a hyperbolic geodesic in the Poincaré ball, Theorem A asserts that there are minimizers of the lift of the Lagrangian system that are a bounded distance away and have a variety of approximate speeds. Theorem B gives the existence of a collection of compact invariant sets of the Euler–Lagrange flow that are semiconjugate to the geodesic flow of a hyperbolic metric. These results can be viewed as a generalization of the Aubry–Mather theory of twist maps and the Hedlund–Morse–Gromov theory of minimal geodesics on closed surfaces and hyperbolic manifolds.

We construct the Green bundles for an energy level without conjugate points of a convex Hamiltonian. In this case we give a formula for the metric entropy of the Liouville measure and prove that the exponential map is a local diffeomorphism. We prove that the Hamiltonian flow is Anosov if and only if the Green bundles are transversal. Using the Clebsch transformation of the index form we prove that if the unique minimizing measure of a generic Lagrangian is supported on a periodic orbit, then it is a hyperbolic periodic orbit.

We also show some examples of differences with the behaviour of a geodesic flow without conjugate points, namely: (non-contact) flows and periodic orbits without invariant transversal bundles, segments without conjugate points but with crossing solutions and non-surjective exponential maps.

We prove the existence of the average linking number, also called the Hopf invariant, for any invariant measure under a differentiable flow on $S^3$ without singularities, which has no periodic orbit of positive measure.