We study families $G(\lambda,\cdot)$ of entire functions that are approximated by a sequence of families $G_n(\lambda,\cdot)$ of entire functions, where $\lambda\in\C$ is a parameter. In order to control the dynamics, the families are assumed to be of the same constant finite type. In this setting we prove the convergence of the hyperbolic components in parameter space as kernels in the sense of Carathéodory.

We study one-dimensional transformations $T$ with indifferent fixed points $p$ and $q$ ($p$ and $q$ are fixed points with $ T^{\prime}(p) = T^{\prime}(q) = 1 $). In the case when $T$ has a Lebesgue equivalent infinite invariant measure, we give the ratio ergodic theorems which describe the limit value of the ratio of the sojourn time of the trajectory $\{T^k(x)\}_{k=0}^n$ in a small neighborhood of $p$ to that in a small neighborhood of $q$.

In this paper we study a generalization of Mayer's result on the Selberg zeta function of $PSL(2, {\Bbb Z})$. Let $\Gamma$ be a cofinite Fuchsian group. We construct a Markov system ${\cal T}_{\Gamma}$ by modifying the Bowen–Series construction of a Markov map $T_{\Gamma}$ associated with $\Gamma$. The Markov system enables us to define transfer operators $L(s)$ for ${\cal T}_{\Gamma}$ so that they determine a meromorphic function taking values with nuclear operators on a nice function space. We show that the Selberg zeta function $Z(s)$ of $\Gamma$ has a determinant representation $Z(s)=\Det(I-L(s))F(s)$, where $\Det(I-L(s))$ is the Fredholm determinant of $L(s)$ and $F(s)$ is a meromorphic function depending only on a finite number of hyperbolic conjugacy classes of $\Gamma$. Combining such a representation and the investigation of the spectral properties of $L(s)$, we can also obtain some analytic information of $Z(s)$.

For any closed hyperbolic manifold of dimension $n \geq 3$, suppose a sequence of $n$-cycles representing the fundamental homology class have norms converging to the Gromov invariant. We show that this sequence must converge to the uniform measure on the space of maximal-volume ideal simplices. As a corollary, we show that for a hyperbolic $n$-manifold $L$ ($n \geq 3$) with totally-geodesic boundary, the Gromov norm of ($L,\partial L$) is strictly greater than the volume of $L$ divided by the maximal volume of an ideal $n$-simplex.

Let $(f,\alpha)$ be the process given by an endomorphism $f$ and by a finite partition $\alpha=\{A\}_{i=1}^{s}$ of a Lebesgue space. Let $E(f,\alpha)$ be the set of densities of absolutely continuous invariant measures for skew products with the base $(f,\alpha)$. We prove that a process $(f,\alpha)$ is Markovian if and only if $E(f,\alpha)\subset \{g: g=\sum_{i=1}^{s} 1_{A_{i}}\otimes g_{i}\}$. If $f$ is the Lasota–Yorke or Misiurewicz type map with the Markovian partition $\alpha$, then $(f,\alpha)$ is quasi-Markovian, i.e. $E(f,\alpha)\subset \{g:\supp g=\bigcup_{i=1}^{s} A_{i}\times B_{i}\}$. Moreover, we give the characterization of quasi-Markovian processes.

Linear mod one transformations are those maps of the unit interval given by $f_{\beta,\alpha}(x)=\beta x+\alpha$ (mod 1), with $\beta>1$ and $0\le\alpha<1$. The lap-counting function is $L_{\beta,\alpha}(z)=\sum_{n=1}^{\infty} L_{n}z^{n}$, where $L_{n}$ essentially counts the number of monotonic pieces of the $n$th iterate $f_{\beta,\alpha}^{n}$. Part I showed that the function $L_{\beta,\alpha}(z)$ is meromorphic in the unit disk $\vert z\vert <1$ and analytic in $\vert z\vert<1/\beta$, and part II showed that the singularities of $L_{\beta,\alpha}(z)$ on the circle $\vert z\vert=1/\beta$ are contained in the set $\{(1/\beta)\exp (2\pi il/N_{\beta,\alpha}):0\le l/N_{\beta,\alpha}\}$, where $N_{\beta,\alpha}$ is the period of the ergodic part of a Markov chain associated to $f_{\beta,\alpha}$. This paper proves that the set of singularities on $\vert z\vert=1/\beta$ is identical to the set $\{(1/\beta)\exp (2\pi il/N_{\beta,\alpha}):0\le l/N_{\beta,\alpha}\}$. Part II showed that $N_{\beta,\alpha}=1$ for $\beta> 2$, and this paper determines $N_{\beta,\alpha}$ in the remaining cases where $1<\beta\le 2$.

Let $G={\rm SL_2({\bf R})}$ (or $G={\rm SO}(n,1)$) act ergodically on a probability space $(X,m)$. We consider the ergodic properties of the flow $(X,m,\{g_t\})$, where $\{g_t\}$ is a Cartan subgroup of $G$. The geodesic flow on a compact Riemann surface is an example of such a flow; here $X=G/\Gamma$ is a transitive $G$-space, $G={\rm SL_2({\bf R})}$ and $\Gamma\subset G$ is a lattice. In this case the flow is Bernoullian.

For the general ergodic $G$-action, the flow $(X,m,\{g_t\})$ is always a $K$-flow, however there are examples in which it is not Bernoullian.

It is an open problem to determine for which maps $f$, any compact invariant set $K$ carries an ergodic invariant measure of the same Hausdorff dimension as $K$. If $f$ is conformal and expanding, then it is a known consequence of the thermodynamic formalism that such measures do exist. (We give a proof here under minimal smoothness assumptions.) If $f$ has the form $f(x_1,x_2)=(f_1(x_1),f_2(x_2))$, where $f_1$ and $f_2$ are conformal and expanding maps satisfying $\inf \vert Df_1\vert\geq\sup\vert Df_2\vert$, then for a large class of invariant sets $K$, we show that ergodic invariant measures of dimension arbitrarily close to the dimension of $K$ do exist. The proof is based on approximating $K$ by self-affine sets.

We study Livsic's problem of finding $\phi$ satisfying $X\phi=\eta$, where $\eta$ is a given function and $X$ is a given Anosov vector field. We show that, if $\phi$ is a continuous solution and $X,\eta$ are analytic, then $\phi$ is analytic. We use the previous result to show that if two low-dimensional Anosov systems are topologically conjugate and the Lyapunov exponents at corresponding periodic points agree, the conjugacy is analytic. Analogous results hold for diffeomorphisms.

We consider $g$-measures for the one-sided shift on a sequence space. We give a condition which guarantees that the $g$-measure is unique and has a Bernoulli natural extension.

The equation $\dot{z} = e^{i \alpha} z + e^{i \varphi} z |z|^2 + b \bar{z}^3$ models a map near a Hopf bifurcation with 1:4 resonance. It is a conjecture by V. I. Arnol'd that this equation contains all versal unfoldings of ${\Bbb Z}_4$-equivariant planar vector fields. We study its bifurcations at $\infty$ and show that the singularities of codimension two unfold versally in a neighborhood. We give an unfolding of the codimension-three singularity for $b=1$, $\varphi=3\pi/2$ and $\alpha=0$ in the system parameters and use numerical methods to study global phenomena to complete the description of the behavior near $\infty$. Our results are evidence in support of the conjecture.

An interval map with only one discontinuity is isomorphic to a rotation of the circle, and has continuous eigenfunctions. What we show here is that for almost every choice of lengths of the intervals, this is the only way an irreducible interval exchange can have a somewhere continuous eigenfunction. We show slightly more, considering certain towers over the interval exchange, showing that outside of a set of choices for interval lengths of measure zero these have a somewhere continuous eigenfunction only if they are isomorphic to either a rotation, or a tower of constant height over an interval exchange.

We show that for translation invariant Markov random fields: (1) the K-property implies a trivial full tail; (2) the Bernoulli property implies Følner independence. The existence of bilaterally deterministic Bernoulli shifts tells us that neither result is true without the Markov assumption (even in one dimension). We also show that for general translation invariant random fields: (3) Følner independence implies a trivial full tail.

We present an optimal result for the regularity of the invariant distributions of an Anosov system in terms of expansion and contraction rates. Compared to earlier results it avoids an ‘infinitesimal loss’ in the Hölder exponent.

Let $ \Phi $ be a $C^2$ codimension one Anosov flow on a compact Riemannian manifold $M$ of dimension greater than three. Verjovsky conjectured that $ \Phi $ admits a global cross-section and we affirm this conjecture when $ \Phi $ is volume preserving in the following two cases: (1) if the sum of the strong stable and strong unstable bundle of $\Phi$ is $ \theta $-Hölder continuous for all $ \theta < 1 $; (2) if the center stable bundle of $ \Phi $ is of class $ C^{1 + \theta} $ for all $ \theta < 1 $. We also show how certain transitive Anosov flows (those whose center stable bundle is $C^1$ and transversely orientable) can be ‘synchronized’, that is, reparametrized so that the strong unstable determinant of the time $t$ map (for all $t$) of the synchronized flow is identically equal to $ e^t $. Several applications of this method are given, including vanishing of the Godbillon–Vey class of the center stable foliation of a codimension one Anosov flow (when $ \dim M > 3 $ and that foliation is $ C^{1 + \theta} $ for all $ \theta < 1 $), and a positive answer to a higher-dimensional analog to Problem 10.4 posed by Hurder and Katok in [HK].

If a non-Möbius inner function on the disk with a fixed point in the open disk has a boundary function which is measure theoretically conjugate to the boundary function of another inner function with fixed point in the open disk, then the inner functions are conformally conjugate.

It is proved that every second countable locally Hausdorff and locally compact continuous groupoid has a Borel set of units that meets every orbit and is what is called ‘lacunary’, a property that implies that the intersection with every orbit is countable.

In this paper we consider the class of surface mappings consisting of those maps which have the least number of fixed points possible among all maps in their homotopy class. When the surface has non-empty boundary, we show that for mappings in this class the index of a fixed point is bounded above by $1$ and below by $2\chi -1$. This generalizes a well known result for pseudo-Anosov homeomorphisms. A proof of a Jiang–Guo type inequality is also given.

The classical criterion for a circle diffeomorphism to be topologically conjugate to an irrational rigid rotation was given by Denjoy [1]. In [5] one of us gave a new criterion. There is an example satisfying Denjoy's bounded variation condition rather than the Zygmund condition of [5], and vice versa. This paper will give the third criterion which is implied by either of the above criteria.

We consider three related classifications of cellular automata: the first is based on the complexity of languages generated by clopen partitions of the state space, i.e. on the complexity of the factor subshifts; the second is based on the concept of equicontinuity and it is a modification of the classification introduced by Gilman [9]. The third one is based on the concept of attractors and it refines the classification introduced by Hurley [16]. We show relations between these classifications and give examples of cellular automata in the intersection classes. In particular, we show that every positively expansive cellular automaton is conjugate to a one-sided subshift of finite type and that every topologically transitive cellular automaton is sensitive to initial conditions. We also construct a cellular automaton with minimal quasi-attractor, whose basin has measure zero, answering a question raised in Hurley [16].