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It is proved that each Gaussian cocycle over a mildly mixing Gaussian transformation is either a Gaussian coboundary or sharply weak mixing. The class of non-singular infinite direct products T of transformations$T_n$,$n\in \mathbb N$, of finite type is studied. It is shown that if$T_n$ is mildly mixing,$n\in \mathbb N$, the sequence of Radon–Nikodym derivatives of$T_n$ is asymptotically translation quasi-invariant and T is conservative then the Maharam extension of T is sharply weak mixing. This technique provides a new approach to the non-singular Gaussian transformations studied recently by Arano, Isono and Marrakchi.
Let
$\phi :X\to X$ be a homeomorphism of a compact metric space X. For any continuous function
$F:X\to \mathbb {R}$ there is a one-parameter group
$\alpha ^{F}$ of automorphisms (or a flow) on the crossed product
$C^*$-algebra
$C(X)\rtimes _{\phi }\mathbb {Z}$ defined such that
$\alpha ^{F}_{t}(fU)=fUe^{-itF}$ when
$f \in C(X)$ and U is the canonical unitary in the construction of the crossed product. In this paper we study the Kubo--Martin--Schwinger (KMS) states for these flows by developing an intimate relation to the ergodic theory of non-singular transformations and show that the structure of KMS states can be very rich and complicated. Our results are complete concerning the set of possible inverse temperatures; in particular, we show that when
$C(X) \rtimes _{\phi } \mathbb Z$ is simple this set is either
$\{0\}$ or the whole line
$\mathbb R$.
We study Kakutani equivalence for products of some special flows over rotations with roof function smooth except a singularity at $0\in \mathbb {T}$. We estimate the Kakutani invariant for products of these flows with different powers of singularities and rotations from a full measure set. As a corollary, we obtain a countable family of pairwise non-Kakutani equivalent products of special flows over rotations.
For random piecewise linear systems T of the interval that are expanding on average we construct explicitly the density functions of absolutely continuous T-invariant measures. If the random system uses only expanding maps our procedure produces all invariant densities of the system. Examples include random tent maps, random W-shaped maps, random $\beta $-transformations and random Lüroth maps with a hole.
For arbitrary closed countable subsets Z of the unit circle examples of topologically mixing operators on Hilbert spaces are given which have a densely spanning set of eigenvectors with unimodular eigenvalues restricted to Z. In particular, these operators cannot be ergodic in the Gaussian sense.
We prove the logarithmic Sarnak conjecture for sequences of subquadratic word growth. In particular, we show that the Liouville function has at least quadratically many sign patterns. We deduce the main theorem from a variant which bounds the correlations between multiplicative functions and sequences with subquadratically many words which occur with positive logarithmic density. This allows us to actually prove that our multiplicative functions do not locally correlate with sequences of subquadratic word growth. We also prove a conditional result which shows that if the (
$\kappa -1$
)-Fourier uniformity conjecture holds then the Liouville function does not correlate with sequences with
$O(n^{t-\varepsilon })$
many words of length n where
$t = \kappa (\kappa +1)/2$
. We prove a variant of the
$1$
-Fourier uniformity conjecture where the frequencies are restricted to any set of box dimension less than
$1$
.
We introduce the notion of balanced strong shift equivalence between square non-negative integer matrices, and show that two finite graphs with no sinks are one-sided eventually conjugate if and only if their adjacency matrices are conjugate to balanced strong shift equivalent matrices. Moreover, we show that such graphs are eventually conjugate if and only if one can be reached by the other via a sequence of out-splits and balanced in-splits, the latter move being a variation of the classical in-split move introduced by Williams in his study of shifts of finite type. We also relate one-sided eventual conjugacies to certain block maps on the finite paths of the graphs. These characterizations emphasize that eventual conjugacy is the one-sided analog of two-sided conjugacy.
We show that the uniform measure-theoretic ergodic decomposition of a countable Borel equivalence relation
$(X, E)$
may be realized as the topological ergodic decomposition of a continuous action of a countable group
$\Gamma \curvearrowright X$
generating E. We then apply this to the study of the cardinal algebra
$\mathcal {K}(E)$
of equidecomposition types of Borel sets with respect to a compressible countable Borel equivalence relation
$(X, E)$
. We also make some general observations regarding quotient topologies on topological ergodic decompositions, with an application to weak equivalence of measure-preserving actions.
We prove the Bernoulli property for determinantal point processes on
$ \mathbb{R}^d $ with translation-invariant kernels. For the determinantal point processes on
$ \mathbb{Z}^d $ with translation-invariant kernels, the Bernoulli property was proved by Lyons and Steif [Stationary determinantal processes: phase multiplicity, bernoullicity, and domination. Duke Math. J.120 (2003), 515–575] and Shirai and Takahashi [Random point fields associated with certain Fredholm determinants II: fermion shifts and their ergodic properties. Ann. Probab.31 (2003), 1533–1564]. We prove its continuum version. For this purpose, we also prove the Bernoulli property for the tree representations of the determinantal point processes.
We adapt techniques developed by Hochman to prove a non-singular ergodic theorem for
$\mathbb {Z}^d$
-actions where the sums are over rectangles with side lengths increasing at arbitrary rates, and in particular are not necessarily balls of a norm. This result is applied to show that the critical dimensions with respect to sequences of such rectangles are invariants of metric isomorphism. These invariants are calculated for the natural action of
$\mathbb {Z}^d$
on a product of d measure spaces.
We prove a theorem describing the limiting fine-scale statistics of orbits of a point in hyperbolic space under the action of a discrete subgroup. Similar results have been proved only in the lattice case with two recent infinite-volume exceptions by Zhang for Apollonian circle packings and certain Schottky groups. Our results hold for general Zariski dense, non-elementary, geometrically finite subgroups in any dimension. Unlike in the lattice case orbits of geometrically finite subgroups do not necessarily equidistribute on the whole boundary of hyperbolic space. But rather they may equidistribute on a fractal subset. Understanding the behavior of these orbits near the boundary is central to Patterson–Sullivan theory and much further work. Our theorem characterises the higher order spatial statistics and thus addresses a very natural question. As a motivating example our work applies to sphere packings (in any dimension) which are invariant under the action of such discrete subgroups. At the end of the paper we show how this statistical characterization can be used to prove convergence of moments and to write down the limiting formula for the two-point correlation function and nearest neighbor distribution. Moreover we establish a formula for the 2 dimensional limiting gap distribution (and cumulative gap distribution) which also applies in the lattice case.
A theorem of Brudno says that the Kolmogorov–Sinai entropy of an ergodic subshift over
$\mathbb {N}$
equals the asymptotic Kolmogorov complexity of almost every word in the subshift. The purpose of this paper is to extend this result to subshifts over computable groups that admit computable regular symmetric Følner monotilings, which we introduce in this work. For every
$d \in \mathbb {N}$
, the groups
$\mathbb {Z}^d$
and
$\mathsf{UT}_{d+1}(\mathbb {Z})$
admit computable regular symmetric Følner monotilings for which the required computing algorithms are provided.
Let
${\mathbf {G}}$
be a semisimple algebraic group over a number field K,
$\mathcal {S}$
a finite set of places of K,
$K_{\mathcal {S}}$
the direct product of the completions
$K_{v}, v \in \mathcal {S}$
, and
${\mathcal O}$
the ring of
$\mathcal {S}$
-integers of K. Let
$G = {\mathbf {G}}(K_{\mathcal {S}})$
,
$\Gamma = {\mathbf {G}}({\mathcal O})$
and
$\pi :G \rightarrow G/\Gamma $
the quotient map. We describe the closures of the locally divergent orbits
${T\pi (g)}$
where T is a maximal
$K_{\mathcal {S}}$
-split torus in G. If
$\# S = 2$
then the closure
$\overline {T\pi (g)}$
is a finite union of T-orbits stratified in terms of parabolic subgroups of
${\mathbf {G}} \times {\mathbf {G}}$
and, consequently,
$\overline {T\pi (g)}$
is homogeneous (i.e.
$\overline {T\pi (g)}= H\pi (g)$
for a subgroup H of G) if and only if
${T\pi (g)}$
is closed. On the other hand, if
$\# \mathcal {S}> 2$
and K is not a
$\mathrm {CM}$
-field then
$\overline {T\pi (g)}$
is homogeneous for
${\mathbf {G}} = \mathbf {SL}_{n}$
and, generally, non-homogeneous but squeezed between closed orbits of two reductive subgroups of equal semisimple K-ranks for
${\mathbf {G}} \neq \mathbf {SL}_{n}$
. As an application, we prove that
$\overline {f({\mathcal O}^{n})} = K_{\mathcal {S}}$
for the class of non-rational locally K-decomposable homogeneous forms
$f \in K_{\mathcal {S}}[x_1, \ldots , x_{n}]$
.
We continue the study of Rokhlin entropy, an isomorphism invariant for probability-measure-preserving (p.m.p.) actions of countablegroups introduced in Part I [B. Seward. Krieger’s finite generator theorem for actions of countable groups I. Invent. Math. 215(1) (2019), 265–310]. In this paper we prove a non-ergodic finite generator theorem and use it to establish sub-additivity and semicontinuity properties of Rokhlin entropy. We also obtain formulas for Rokhlin entropy in terms of ergodic decompositions and inverse limits. Finally, we clarify the relationship between Rokhlin entropy, sofic entropy, and classical Kolmogorov–Sinai entropy. In particular, using Rokhlin entropy we give a new proof of the fact that ergodic actions with positive sofic entropy have finite stabilizers.
We prove several general conditional convergence results on ergodic averages for horocycle and geodesic subgroups of any continuous
$\operatorname {SL}(2, \mathbb {R})$
-action on a locally compact space. These results are motivated by theorems of Eskin, Mirzakhani and Mohammadi on the
$\operatorname {SL}(2, \mathbb {R})$
-action on the moduli space of Abelian differentials. By our argument we can derive from these theorems an improved version of the ‘weak convergence’ of push-forwards of horocycle measures under the geodesic flow and a short proof of weaker versions of theorems of Chaika and Eskin on Birkhoff genericity and Oseledets regularity in almost all directions for the Teichmüller geodesic flow.
We obtain large and moderate deviation estimates for both sequential and random compositions of intermittent maps. We also address the question of whether or not centering is necessary for the quenched central limit theorems obtained by Nicol, Török and Vaienti [Central limit theorems for sequential and random intermittent dynamical systems. Ergod. Th. & Dynam. Sys.38(3) (2018), 1127–1153] for random dynamical systems comprising intermittent maps. Using recent work of Abdelkader and Aimino [On the quenched central limit theorem for random dynamical systems. J. Phys. A 49(24) (2016), 244002] and Hella and Stenlund [Quenched normal approximation for random sequences of transformations. J. Stat. Phys.178(1) (2020), 1–37] we extend the results of Nicol, Török and Vaienti on quenched central limit theorems for centered observables over random compositions of intermittent maps: first by enlarging the parameter range over which the quenched central limit theorem holds; and second by showing that the variance in the quenched central limit theorem is almost surely constant (and the same as the variance of the annealed central limit theorem) and that centering is needed to obtain this quenched central limit theorem.
We obtain estimates on the uniform convergence rate of the Birkhoff average of a continuous observable over torus translations and affine skew product toral transformations. The convergence rate depends explicitly on the modulus of continuity of the observable and on the arithmetic properties of the frequency defining the transformation. Furthermore, we show that for the one-dimensional torus translation, these estimates are nearly optimal.
A measure on a locally compact group is said to be spread out if one of its convolution powers is not singular with respect to Haar measure. Using Markov chain theory, we conduct a detailed analysis of random walks on homogeneous spaces with spread out increment distribution. For finite volume spaces, we arrive at a complete picture of the asymptotics of the n-step distributions: they equidistribute towards Haar measure, often exponentially fast and locally uniformly in the starting position. In addition, many classical limit theorems are shown to hold. In the infinite volume case, we prove recurrence and a ratio limit theorem for symmetric spread out random walks on homogeneous spaces of at most quadratic growth. This settles one direction in a long-standing conjecture.
We show that for good measures, the set of homeomorphisms of Cantor space which preserve that measure and which have no invariant clopen sets contains a residual set of homeomorphisms which are uniquely ergodic. Additionally, we show that for refinable Bernoulli trial measures, the same set of homeomorphisms contains a residual set of homeomorphisms which admit only finitely many ergodic measures.
This article investigates the long-time behavior of conservative affine processes on the cone of symmetric positive semidefinite
$d\times d$
matrices. In particular, for conservative and subcritical affine processes we show that a finite
$\log$
-moment of the state-independent jump measure is sufficient for the existence of a unique limit distribution. Moreover, we study the convergence rate of the underlying transition kernel to the limit distribution: first, in a specific metric induced by the Laplace transform, and second, in the Wasserstein distance under a first moment assumption imposed on the state-independent jump measure and an additional condition on the diffusion parameter.