One may often decompose the domain of a topologically transitivemap into finitely many regular closed pieces with nowhere denseoverlap in such a way that these pieces map into one another in aperiodic fashion. We call decompositions of this kind regularperiodic decompositions and refer to the number of pieces asthe length of the decomposition. If $f$ is topologically transitivebut $f^{n}$ is not, then $f$ has a regular periodic decomposition ofsome length dividing $n$. Although a decomposition of a givenlength is unique, a map may have many decompositions ofdifferent lengths. The set of lengths of decompositions of a givenmap is an ideal in the lattice of natural numbers ordered bydivisibility, which we call the decomposition ideal of $f$. Everyideal in this lattice arises as a decomposition ideal of some map.Decomposition ideals of Cartesian products of transitive mapsare discussed and used to develop various examples. Results areobtained concerning the implications of local connectedness fordecompositions. We conclude with a comprehensive analysis ofthe possible decomposition ideals for maps on 1-manifolds.

Given a ${\cal C}^{1+\gamma}$ hyperbolic Cantor set$C$, we study the sequence $C_{n,x}$ of Cantor subsets which nest down towarda point $x$ in $C$. We show that $C_{n,x}$ is asymptotically equal to anergodic Cantor set valued process. The values of this process, calledlimit sets, are indexed by a Hölder continuous set-valuedfunction definedon Sullivan's dual Cantor set. We show the limit sets are themselves ${\calC}^{k+\gamma},{\cal C}^\infty$ or ${\cal C}^\omega$ hyperbolic Cantor sets,with the highest degree of smoothness which occurs in the ${\calC}^{1+\gamma}$ conjugacy class of $C$. The proof of this leads to thefollowing rigidity theorem: if two ${\cal C}^{k+\gamma},{\cal C}^\infty$ or${\cal C}^\omega$ hyperbolic Cantor sets are ${\cal C}^1$ conjugate, then theconjugacy (with a different extension) is in fact already ${\calC}^{k+\gamma},{\cal C}^\infty$ or ${\cal C}^\omega$. Within one ${\calC}^{1+\gamma}$ conjugacy class, each smoothness class is a Banach manifold,which is acted on by the semigroup given by rescaling subintervals.Smoothness classes nest down, and contained in the intersection of them allis a compact set which is the attractor for the semigroup: the collection oflimit sets. Convergence is exponentially fast, in the ${\cal C}^1$ norm.

We propose a new classification of periodic orbits of interval maps viaover-rotation pairs. We prove a theorem for them similar to theSharkovskii theorem.

We extend the theory of rotation vectors tohomeomorphisms of the two-dimensional torus that are homotopic to a Dehntwist. We define a one-dimensional rotation number and recreate thetheory of the homotopic case to the identity case. We prove that if such amap is area preserving and has mean rotation number zero, then it musthave a fixed point. We prove that the rotation set is a compactinterval, and that if the rotation interval contains two distinctnumbers, then for any rational number in the rotation set there exists aperiodic point with that rotation number. Finally, we prove that anyinterval with rational endpoints can be realized as therotation set of a map homotopic to a Dehn twist.

The two-dimensional analogue of the Sharkovski order on periodsfor maps of the interval restricts to a partial order on essentialpseudo-Anosov conjugacy classes in the mapping class group ofthe $n$-times punctured disk. In this paper we give an explicitdescription of this restricted partial order in the case when $n=3$.

We construct a universal Cuntz–Krieger algebra ${\cal {AO}}_A$, whichis isomorphic to the usual Cuntz–Krieger algebra ${\cal O}_A$ when $A$satisfiescondition $(I)$ of Cuntz and Krieger. The Cuntz classification of ideals in${\cal O}_A$ when $A$ satisfies condition $(II)$ extends to aclassification of the gauge invariant ideals in ${\cal {AO}}_A$. We use thistodescribe the topology on the primitive ideal space of ${\cal {AO}}_A$.

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 infiniteinvariant measure, we give the ratio ergodic theorems which describe thelimit 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 smallneighborhood of $q$.

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

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 showthat, if $\phi$ is a continuous solution and $X,\eta$ are analytic, then$\phi$ is analytic. We use the previous result to show that if twolow-dimensional Anosov systems are topologically conjugate and the Lyapunovexponents at corresponding periodic points agree, the conjugacy is analytic.Analogous results hold for diffeomorphisms.

If a non-Möbius inner function on the disk witha fixed point in theopen disk has a boundary function which is measuretheoretically conjugate to the boundary function of anotherinner function with fixed point in the open disk, then theinner functions are conformally conjugate.

We construct a minimal foliation of thin type which isnot uniquely ergodic. The notion of thin type relates to Rips'classification of foliations on 2-complexes.

More than 30 years ago R\'enyi [1] introduced the representations ofreal numbers with an arbitrary base $\beta > 1$ as a generalization of the$p$-adic representations. One of the most studied problems in this field isthe link between expansions to base $\beta$ and ergodic properties of thecorresponding $\beta$-shift.

In this paper we will follow the bibliography of Blanchard [2] andgivean affirmative answer to a question on the size of the set of real numbers$\beta$ having complicated symbolic dynamics of their $\beta$-shifts.

This paper investigates dynamical systems arising from theaction bytranslations on the orbit closures of self-similar andself-affine tilingsof ${\Bbb R}^d$. The main focus is on spectral properties of suchsystemswhich are shown to be uniquely ergodic. We establishcriteria forweak mixing and pure discrete spectrum for wide classes ofsuch systems.They are applied to a number of examples which includetilings withpolygonal and fractal tile boundaries; systems with purediscrete,continuous and mixed spectrum.

We study the Hausdorff dimension of invariantsets for expanding maps and that of hyperbolic sets on unstablemanifolds. Upper bounds for the Hausdorff dimension are given in termsof topological pressure, or topological entropy and Lyapunovexponents.