On the geometry and regularity of invariant sets of piecewise-affine automorphisms on the Euclidean space

In this paper, we derive geometric and analytic properties of invariant sets, including orbit closures, of a large class of piecewise-affine maps $T$ on $\mathbb{R}^{d}$. We assume that (i) $T$ consists of finitely many affine maps defined on a Borel measurable partition of $\mathbb{R}^{d}$, (ii) there is a lattice $\mathscr{L}\subset \mathbb{R}^{d}$ that contains all of the mutual differences of the translation vectors of these affine maps, and (iii) all of the affine maps have the same linear part that is an automorphism of $\mathscr{L}$. We prove that finite-volume invariant sets of such piecewise-affine maps always consist of translational tiles relative to this lattice, up to some multiplicity. When the partition is Jordan measurable, we show that closures of bounded orbits of $T$ are invariant and yield Jordan measurable tiles, again up to some multiplicity. In the latter case, we show that compact $T$-invariant sets also consist of Jordan measurable tiles. We then utilize these results to quantify the rate of convergence of ergodic averages for $T$ in the case of bounded single tiles.

A family of densities derived from the three-parameter Dirichlet process

The traditional Dirichlet process is characterized by its distribution on a measurable partition of the state space - namely, the Dirichlet distribution. In this paper, we consider a generalization of the Dirichlet process and the family of multivariate distributions it induces, with particular attention to a special case where the multivariate density function is tractable.

A family of densities derived from the three-parameter Dirichlet process

The traditional Dirichlet process is characterized by its distribution on a measurable partition of the state space - namely, the Dirichlet distribution. In this paper, we consider a generalization of the Dirichlet process and the family of multivariate distributions it induces, with particular attention to a special case where the multivariate density function is tractable.

On finitely Bernoulli one-sided Markov shifts and their cofiltrations

The problem under consideration is: when is a Markov endomorphism (one-sided shift) $T=T_P$ with transition matrix $P$, isomorphic to a Bernoulli endomorphism $\tilde{T}_\rho$ with an appropriate stationary vector $\rho=\{\rho_i\}_{i\in I}$? An obvious necessary condition is that there exists an independent complement $\delta$ of the measurable partition $T^{-1}\varepsilon$ with $\distr \delta = \rho$. In this case the cofiltration (decreasing sequence of measurable partitions) $\xi(T)=\{T^{-n}\varepsilon\}^{\infty}_{n=1}$ generated by $T$ is finitely isomorphic to the standard Bernoulli cofiltration $\xi(\tilde{T}_{\rho})=\{\tilde{T}^{-n}_{\rho}\varepsilon\}^{\infty}_{n=1}$ and $T$ is called finitely $\rho$-Bernoulli.We show that every ergodic Markov endomorphism $T_P$, which is finitely Bernoulli, can be represented as a skew product over $\tilde{T}_{\rho}$ with $d$-point fibres $(d\in \mathbb{N})$. We compute the minimal $d=d(T_P)$ in these skew-product representations by means of the transition matrix, and obtain necessary and sufficient conditions under which $d(T_P)=1$, i.e. $T_P$ is isomorphic to $\tilde{T}_{\rho}$. The cofiltration $\xi(T)$ of any finitely Bernoulli ergodic Markov endomorphism $T=T_P$ is represented as a $d$-point extension of the standard cofiltration $\xi(\tilde{T}_{\rho})$, and we show that the minimal $d=d_{\xi}(T)$ in these extensions is equal to $d(T)$. In particular, $d(T)=1\Longleftrightarrow d_{\xi}(T)=1$, that is, a Markov endomorphism $T$ is isomorphic to $\tilde{T}_{\rho}$ iff $\xi(T)$ is isomorphic to the Bernoulli cofiltration $\xi(\tilde{T}_{\rho})$.

Normalizers of finite multiplicity nests

We show that every continuous nest of bounded multiplicity is unitarily equivalent to itself in a non-trivial way. Along the way, it is shown that no finite (measurable) partition of the unit interval can separate absolutely continuous homeomorphisms.

On a theorem of Dvoretsky, Wald, and Wolfowitz concerning Liapounov Measures

Let ω be a non-empty set, ℱ a Boolean σ-algebra of subsets of Ω, k a natural number, and let m:ℱ→ℝk be a non-atomic vector measure. Then, by the celebrated theorem of Liapounov [11], the range m[3F] = {m(A): A ε ℱ3F} of m is a compact convex subset of ℝk. This theorem has been generalized in a number of ways. For example Kingman and Robertson [8] and Knowles [9] have shown that, under appropriate conditions, results in the same spirit can be proved for measures taking their values in infinite-dimensional vector spaces. Another type of generalization was obtained by Dvoretsky, Wald and Wolfowitz [6,7]. What they do is to take m as above together with a natural number n≥ 1. They then consider the set Knof all vectorswhere (A1 A2,…, An) is an ordered ℱ-measurable partition of Ω (i.e. a partition whose terms A, all belong to ℱ). They prove in [6] that Kn is a compact convex subset of ℝnk and moreover that Kn is equal to the set of all vectors of the formwhere (ϕ1, ϕ2…, ϕn) is an ℱ-measurable partition of unity; i.e. it is an n-tuple of non-negative ϕr on Ω such thatLiapounov's theorem can be obtained as a corollary of this result by taking n= 2.