arXiv subjects

Klaus Schmidt

Publications and source records attributed to Klaus Schmidt.

24 records · Page 2Linked to original sources

arXiv2006

Expansive algebraic actions of discrete residually finite amenable groups and their entropy

We prove an entropy formula for certain expansive actions of a countable discrete residually finite group $Γ$ by automorphisms of compact abelian groups in terms of Fuglede-Kadison determinants. This extends an earlier result proved by the first author under somewhat more restrictive conditions. The main tools for this generalization are a representation of the $Γ$-action by means of a `fundamental homoclinic point', and the description of entropy in terms of the renormalized logarithmic growth-rate of the set of $Γ_n$-fixed points, where $(Γ_n, n\ge1)$ is a decreasing sequence of finite index normal subgroups of $Γ$ with trivial intersection.

math.DS↗

arXiv2006

Recurrence of cocycles and stationary random walks

We survey distributional properties of $\mathbb{R}^d$-valued cocycles of finite measure preserving ergodic transformations (or, equivalently, of stationary random walks in $\mathbb{R}^d$) which determine recurrence or transience.

math.DS↗

arXiv2005

Symbolic representations of nonexpansive group automorphisms

If $α$ is an irreducible nonexpansive ergodic automorphism of a compact abelian group $X$ (such as an irreducible nonhyperbolic ergodic toral automorphism), then $α$ has no finite or infinite state Markov partitions, and there are no nontrivial continuous embeddings of Markov shifts in $X$. In spite of this we are able to construct a symbolic space $V$ and a class of shift-invariant probability measures on $V$ each of which corresponds to an $α$-invariant probability measure on $X$. Moreover, every $α$-invariant probability measure on $X$ arises essentially in this way. The last part of the paper deals with the connection between the two-sided beta-shift $V_β$ arising from a Salem number $β$ and the nonhyperbolic ergodic toral automorphism $α$ arising from the companion matrix of the minimal polynomial of $β$, and establishes an entropy-preserving correspondence between a class of shift-invariant probability measures on $V_β$ and certain $α$-invariant probability measures on $X$. This correspondence is much weaker than, but still quite closely modelled on, the connection between the two-sided beta-shifts defined by Pisot numbers and the corresponding hyperbolic ergodic toral automorphisms.

math.DS↗

arXiv2003

Invariant sets and measures of nonexpansive group automorphisms

We prove that the restriction of a probability measure invariant under a nonhyperbolic, ergodic and totally irreducible automorphism of a compact connected abelian group to the leaves of the central foliation is severely restricted. We also prove a topological analogue of this result: the intersection of every proper closed invariant subset with each central leaf is compact.

math.DS↗

arXiv2000

Rigidity of measurable structure for Z^d-actions by automorphisms of a torus

We show that for certain classes of actions of Z^d, d >= 2, by automorphisms of the torus any measurable conjugacy has to be affine, hence measurable conjugacy implies algebraic conjugacy; similarly any measurable factor is algebraic, and algebraic and affine centralizers provide invariants of measurable conjugacy. Using the algebraic machinery of dual modules and information about class numbers of algebraic number fields we consruct various examples of Z^d-actions by Bernoulli automorphisms whose measurable orbit structure is rigid, including actions which are weakly isomorphic but not isomorphic. We show that the structure of the centralizer for these actions may or may not serve as a distinguishing measure-theoretic invariant.

math.DS↗

arXiv1996

Symmetric Gibbs measures

We prove that certain Gibbs measures on subshifts of finite type are nonsingular and ergodic for certain countable equivalence relations, including the orbit relation of the adic transformation (the same as equality after a permutation of finitely many coordinates). The relations we consider are defined by cocycles taking values in groups, including some nonabelian ones. This generalizes (half of) the identification of the invariant ergodic probability measures for the Pascal adic transformation as exactly the Bernoulli measures---a version of de Finetti's Theorem. Generalizing the other half, we characterize the measures on subshifts of finite type that are invariant under both the adic and the shift as the Gibbs measures whose potential functions depend on only a single coordinate. There are connections with and implications for exchangeability, ratio limit theorems for transient Markov chains, interval splitting procedures, `canonical' Gibbs states, and the triviality of remote sigma-fields finer than the usual tail field.

math.DS↗