For S-unimodal maps $f$, we study equilibrium states maximizing the free energies $F_t(\mu) := h(\mu) - t\int \log|f'|\,d\mu$ and the pressure function $P(t):=\sup_\mu F_t(\mu)$. It is shown that if $f$ is uniformly hyperbolic on periodic orbits, then $P(t)$ is analytic for $t\approx 1$. On the other hand, examples are given where no equilibrium states exist, where equilibrium states are not unique and where the notions of equilibrium state for $t=1$ and of observable measure do not coincide.

In this paper we continue to explore the applications of the connections between singular Riemannian geometry and billiard systems that were first used in [6] to prove estimates on the number of collisions in non-degenerate semi-dispersing billiards.

In this paper we show that the topological entropy of a compact non-degenerate semi-dispersing billiard on any manifold of non-positive sectional curvature is finite. Also, we prove exponential estimates on the number of periodic points (for the first return map to the boundary of a simple-connected billiard table) and the number of periodic trajectories (for the billiard flow). In \S5 we prove some estimates for the topological entropy of Lorentz gas.

In this paper we show that manifolds without conjugate points exhibit rigidity phenomena similar to that studied in [BGS, Section I.5]. The main theorem is that if $X$ is a complete, simply connected Riemannian manifold without conjugate points, and $M=X\times R$ is given the Riemannian product metric $g$, then any metric without conjugate points on $M$ which agrees with $g$ outside a compact set is isometric to $g$.

In this paper we show that some nonsimply connected manifolds without conjugate points exhibit rigidity phenomena similar to that studied in [BGS, section I.5]. This is a companion paper to [Cr-Kl1] that deals with the simply connected case. In particular, we show that one cannot make a nontrivial, compactly supported, change to a complete flat metric without introducing conjugate points.

A conjugacy between two uniformly asymptotic affine circle expanding maps is symmetric if they have the same eigenvalues along corresponding periodic points.

Let $\Gamma$ denote a word-hyperbolic group, and let $S=S^{-1}$ denote a finite symmetric set of generators. Let $S_n = \{w: |w| = n \}$ denote the sphere of radius $n$, where $|\cdot|$ denotes the word length on $\Gamma$ induced by $S$. Define $\s_n \deq (1/{\#S_n}) \sum_{w \in S_n}w$, and $\mu_n=(1/(n+1))\sum_{k=0}^n \s_k$. Let $(X,{\cal B},m)$ be a probability space on which $\Gamma$ acts ergodically by measure-preserving transformations. We prove a strong maximal inequality in $L^2$ for the maximal operator $f_{\mu}^*=\sup_{n\ge 0}|\mu_n f(x)|$. The maximal inequality is applied to prove a pointwise ergodic theorem in $L^2$ for exponentially mixing actions of $\Gamma$ of the following form: $\mu_n f(x) \to \int_X f\,dm$ almost everywhere and in the $L^2$-norm, for every $f\in L^2(X)$. As a corollary, for a uniform lattice $\Gamma \subset G$, where $G$ is a simple Lie group of real rank one, we obtain a pointwise ergodic theorem for the action of $\Gamma$ on an arbitrary ergodic $G$-space. In particular, this result holds when $X=G/\Lambda$ is a compact homogeneous space, and yields an equidistribution result for sets of lattice points of the form $\Gamma g$, for almost every $g\in G$.

We study automorphisms of the CAR-algebra obtained from binary shifts. We consider cases when the $C^\ast$-dynamical system is asymptotically abelian, is proximally asymptotically abelian, is an entropic $K$-system or has completely positive entropy. The entropy is computed in several cases.

A sequence of random variables, $Y_0,Y_1,Y_2,\ldots\,$, is called standard if there exists a one-sided isomorphism between it and a sequence of independent random variables. In this paper it is demonstrated that the sequence arising from the past of the $T,T^{-1}$ map is not standard.

We study spectral properties of the flow $\dot x =1/F(x,y)$, $\dot y = 1/\lambda F(x,y)$ on the 2-torus. We show that, in general, the speed of approximation in cyclic approximation gives an upper bound on the Hausdorff dimension of the supports of spectral measures. We use this to prove that for generic pairs $(F,\lambda)$ the spectrum of the flow on the torus is singular continuous with all spectral measures supported on sets of zero Hausdorff dimension.

In this paper we establish a variety of square function inequalities and study other operators which measure the oscillation of a sequence of ergodic averages. These results imply the pointwise ergodic theorem and give additional information such as control of the number of upcrossings of the ergodic averages. Related results for differentiation and for the connection between differentiation operators and the dyadic martingale are also established.

Given a real number $\beta > 1$, we construct a simple purely infinite $C^*$-algebra ${\cal O}_{\beta}$ as a $C^*$-algebra arising from the $\beta$-subshift in the symbolic dynamics. The $C^*$-algebras $\{{\cal O}_{\beta} \}_{1<\beta \in {\Bbb R}}$ interpolate between the Cuntz algebras $\{{\cal O}_n\}_{1 < n \in {\Bbb N}}$. The K-groups for the $C^*$-algebras ${\cal O}_{\beta}$, $1 < \beta \in {\Bbb R}$, are computed so that they are completely classified up to isomorphism. We prove that the KMS-state for the gauge action on ${\cal O}_{\beta}$ is unique at the inverse temperature $\log \beta$, which is the topological entropy for the $\beta$-shift. Moreover, ${\cal O}_{\beta}$ is realized to be a universal $C^*$-algebra generated by $n-1=[\beta]$ isometries and one partial isometry with mutually orthogonal ranges and a certain relation coming from the sequence of $\beta$-expansion of $1$.

Le mélange faible topologique de presque tout échange d'intervalles différent d'une rotation a été démontré, en 1990, par Nogueira et Rudolph. On généralise ici ce résultat en considérant des flots suspensions au-dessus d'échanges d'intervalles : on montre que pour presque toute forme différentielle quadratique orientable sur une surface de genre ${\ge}2$, le flot vertical associé est topologiquement faiblement mélangeant ; le cas du tore permettant, lui, d'obtenir des fonctions propres continues naturelles par projection sur l'une des coordonnées.

We study cohomology-free (c.f.) diffeomorphisms of the torus $T^n$. A diffeomorphism is c.f. if every smooth function $f$ on $T^n$ is cohomologous to a constant $f_0$, i.e. there exists a $C^{\infty}$ function $h$ so that $h-h\circ\varphi=f-f_0$. We show that the only c.f. diffeomorphisms of $T^n$, $1\le n\le3$, are the smooth conjugations of Diophantine translations. For $n=4$, we prove the same result for c.f. orientation-preserving diffeomorphisms.

Let $F:{\Bbb R}^2 \to {\Bbb R}^2$ be a mapping that is analytic and area preserving. If $F\neq \hbox{\it identity}$, then every stable fixed point is isolated.

This result can be applied to prove that the number of stable periodic solutions of a fixed period of certain Hamiltonian systems is finite.

We define, following Veech, the Fuchsian group $\Gamma(P)$ of a rational polygon $P$. If $P$ is simply-connected, then ‘rational’ is equivalent to the condition that all interior angles of $P$ be rational multiples of $\pi$. Should it happen that $\Gamma(P)$ has finite covolume in $\mathop{\rm PSL}\nolimits (2, {\Bbb R})$ (and is thus a {\it lattice}), then a theorem of Veech states that every billiard path in $P$ is either finite or uniformly distributed in $P$.

We consider the Fuchsian groups of various rational triangles. First, we calculate explicitly the Fuchsian groups of a new sequence of triangles, and discover they are lattices. Interestingly, the lattices found are not commensurable with those previously known. We then demonstrate a class of triangles whose Fuchsian groups are {\it not\/} lattices. These are the first examples of such triangles. Finally, we end by showing how one may specify algebraically, i.e. by an explicit polynomial in two variables, the Riemann surfaces and holomorphic one-forms that are associated to a simply-connected rational polygon. Previously, these surfaces were known by their geometric description. As an example, we show a connection between the billiard in a regular polygon and the well-known Fermat curves of the algebraic equation $x^n + y^n = 1$.