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Under certain assumptions on CAT(0) spaces, we show that the geodesic flow is topologically mixing. In particular, the Bowen–Margulis’ measure finiteness assumption used by Ricks [Flat strips, Bowen–Margulis measures, and mixing of the geodesic flow for rank one CAT(0) spaces. Ergod. Th. & Dynam. Sys.37 (2017), 939–970] is removed. We also construct examples of CAT(0) spaces that do not admit finite Bowen–Margulis measure.
We obtain a central limit theorem, local limit theorems and renewal theorems for stationary processes generated by skew product maps
$T(\unicode[STIX]{x1D714},x)=(\unicode[STIX]{x1D703}\unicode[STIX]{x1D714},T_{\unicode[STIX]{x1D714}}x)$
together with a
$T$
-invariant measure whose base map
$\unicode[STIX]{x1D703}$
satisfies certain topological and mixing conditions and the maps
$T_{\unicode[STIX]{x1D714}}$
on the fibers are certain non-singular distance-expanding maps. Our results hold true when
$\unicode[STIX]{x1D703}$
is either a sufficiently fast mixing Markov shift with positive transition densities or a (non-uniform) Young tower with at least one periodic point and polynomial tails. The proofs are based on the random complex Ruelle–Perron–Frobenius theorem from Hafouta and Kifer [Nonconventional Limit Theorems and Random Dynamics. World Scientific, Singapore, 2018] applied with appropriate random transfer operators generated by
$T_{\unicode[STIX]{x1D714}}$
, together with certain regularity assumptions (as functions of
$\unicode[STIX]{x1D714}$
) of these operators. Limit theorems for deterministic processes whose distributions on the fibers are generated by Markov chains with transition operators satisfying a random version of the Doeblin condition are also obtained. The main innovation in this paper is that the results hold true even though the spectral theory used in Aimino, Nicol and Vaienti [Annealed and quenched limit theorems for random expanding dynamical systems. Probab. Theory Related Fields162 (2015), 233–274] does not seem to be applicable, and the dual of the Koopman operator of
$T$
(with respect to the invariant measure) does not seem to have a spectral gap.
We define a natural topology on the collection of (equivalence classes up to scaling of) locally finite measures on a homogeneous space and prove that in this topology, pushforwards of certain infinite-volume orbits equidistribute in the ambient space. As an application of our results we prove an asymptotic formula for the number of integral points in a ball on some varieties as the radius goes to infinity.
We show that dynamical systems with
$\unicode[STIX]{x1D719}$
-mixing measures have local escape rates which are exponential with rate
$1$
at non-periodic points and equal to the extremal index at periodic points. We apply this result to equilibrium states on subshifts of finite type, Gibbs–Markov systems, expanding interval maps, Gibbs states on conformal repellers, and more generally to Young towers, and by extension to all systems that can be modeled by a Young tower.
An isotopic to the identity map of the 2-torus, that has zero rotation vector with respect to an invariant ergodic probability measure, has a fixed point by a theorem of Franks. We give a version of this result for nilpotent subgroups of isotopic to the identity diffeomorphisms of the 2-torus. In such a context we guarantee the existence of global fixed points for nilpotent groups of irrotational diffeomorphisms. In particular, we show that the derived group of a nilpotent group of isotopic to the identity diffeomorphisms of the 2-torus has a global fixed point.
We investigate the growth rate of the Birkhoff sums
$S_{n,\unicode[STIX]{x1D6FC}}f(x)=\sum _{k=0}^{n-1}f(x+k\unicode[STIX]{x1D6FC})$
, where
$f$
is a continuous function with zero mean defined on the unit circle
$\mathbb{T}$
and
$(\unicode[STIX]{x1D6FC},x)$
is a ‘typical’ element of
$\mathbb{T}^{2}$
. The answer depends on the meaning given to the word ‘typical’. Part of the work will be done in a more general context.
We study sets of measure-preserving transformations on Lebesgue spaces with continuous measures taking into account extreme scales of variations of weak mixing. It is shown that the generic dynamical behaviour depends on subsequences of time going to infinity. We also present corresponding generic sets of (probability) invariant measures with respect to topological shifts over finite alphabets and Axiom A diffeomorphisms over topologically mixing basic sets.
Eagleson’s theorem asserts that, given a probability-preserving map, if renormalized Birkhoff sums of a function converge in distribution, then they also converge with respect to any probability measure which is absolutely continuous with respect to the invariant one. We prove a version of this result for almost sure limit theorems, extending results of Korepanov. We also prove a version of this result, in mixing systems, when one imposes a conditioning both at time
$0$
and at time
$n$
.
We give an integrability criterion on a real-valued non-increasing function
$\unicode[STIX]{x1D713}$
guaranteeing that for almost all (or almost no) pairs
$(A,\mathbf{b})$
, where
$A$
is a real
$m\times n$
matrix and
$\mathbf{b}\in \mathbb{R}^{m}$
, the system
is solvable in
$\mathbf{p}\in \mathbb{Z}^{m}$
,
$\mathbf{q}\in \mathbb{Z}^{n}$
for all sufficiently large
$T$
. The proof consists of a reduction to a shrinking target problem on the space of grids in
$\mathbb{R}^{m+n}$
. We also comment on the homogeneous counterpart to this problem, whose
$m=n=1$
case was recently solved, but whose general case remains open.
We introduce and study skew product Smale endomorphisms over finitely irreducible shifts with countable alphabets. This case is different from the one with finite alphabets and we develop new methods. In the conformal context we prove that almost all conditional measures of equilibrium states of summable Hölder continuous potentials are exact dimensional and their dimension is equal to the ratio of (global) entropy and Lyapunov exponent. We show that the exact dimensionality of conditional measures on fibers implies global exact dimensionality of the original measure. We then study equilibrium states for skew products over expanding Markov–Rényi transformations and settle the question of exact dimensionality of such measures. We apply our results to skew products over the continued fraction transformation. This allows us to extend and improve the Doeblin–Lenstra conjecture on Diophantine approximation coefficients to a larger class of measures and irrational numbers.
We consider random walks on the mapping class group that have finite first moment with respect to the word metric, whose support generates a non-elementary subgroup and contains a pseudo-Anosov map whose invariant Teichmüller geodesic is in the principal stratum of quadratic differentials. We show that a Teichmüller geodesic typical with respect to the harmonic measure for such random walks, is recurrent to the thick part of the principal stratum. As a consequence, the vertical foliation of such a random Teichmüller geodesic has no saddle connections.
A one-sided shift of finite type
$(\mathsf{X}_{A},\unicode[STIX]{x1D70E}_{A})$
determines on the one hand a Cuntz–Krieger algebra
${\mathcal{O}}_{A}$
with a distinguished abelian subalgebra
${\mathcal{D}}_{A}$
and a certain completely positive map
$\unicode[STIX]{x1D70F}_{A}$
on
${\mathcal{O}}_{A}$
. On the other hand,
$(\mathsf{X}_{A},\unicode[STIX]{x1D70E}_{A})$
determines a groupoid
${\mathcal{G}}_{A}$
together with a certain homomorphism
$\unicode[STIX]{x1D716}_{A}$
on
${\mathcal{G}}_{A}$
. We show that each of these two sets of data completely characterizes the one-sided conjugacy class of
$\mathsf{X}_{A}$
. This strengthens a result of Cuntz and Krieger. We also exhibit an example of two irreducible shifts of finite type which are eventually conjugate but not conjugate. This provides a negative answer to a question of Matsumoto of whether eventual conjugacy implies conjugacy.
We prove simplicity of all intermediate
$C^{\ast }$
-algebras
$C_{r}^{\ast }(\unicode[STIX]{x1D6E4})\subseteq {\mathcal{B}}\subseteq \unicode[STIX]{x1D6E4}\ltimes _{r}C(X)$
in the case of minimal actions of
$C^{\ast }$
-simple groups
$\unicode[STIX]{x1D6E4}$
on compact spaces
$X$
. For this, we use the notion of stationary states, recently introduced by Hartman and Kalantar [Stationary
$C^{\ast }$
-dynamical systems. Preprint, 2017, arXiv:1712.10133]. We show that the Powers’ averaging property holds for the reduced crossed product
$\unicode[STIX]{x1D6E4}\ltimes _{r}{\mathcal{A}}$
for any action
$\unicode[STIX]{x1D6E4}\curvearrowright {\mathcal{A}}$
of a
$C^{\ast }$
-simple group
$\unicode[STIX]{x1D6E4}$
on a unital
$C^{\ast }$
-algebra
${\mathcal{A}}$
, and use it to prove a one-to-one correspondence between stationary states on
${\mathcal{A}}$
and those on
$\unicode[STIX]{x1D6E4}\ltimes _{r}{\mathcal{A}}$
.
In this paper we consider the algebraic crossed product
${\mathcal{A}}:=C_{K}(X)\rtimes _{T}\mathbb{Z}$
induced by a homeomorphism
$T$
on the Cantor set
$X$
, where
$K$
is an arbitrary field with involution and
$C_{K}(X)$
denotes the
$K$
-algebra of locally constant
$K$
-valued functions on
$X$
. We investigate the possible Sylvester matrix rank functions that one can construct on
${\mathcal{A}}$
by means of full ergodic
$T$
-invariant probability measures
$\unicode[STIX]{x1D707}$
on
$X$
. To do so, we present a general construction of an approximating sequence of
$\ast$
-subalgebras
${\mathcal{A}}_{n}$
which are embeddable into a (possibly infinite) product of matrix algebras over
$K$
. This enables us to obtain a specific embedding of the whole
$\ast$
-algebra
${\mathcal{A}}$
into
${\mathcal{M}}_{K}$
, the well-known von Neumann continuous factor over
$K$
, thus obtaining a Sylvester matrix rank function on
${\mathcal{A}}$
by restricting the unique one defined on
${\mathcal{M}}_{K}$
. This process gives a way to obtain a Sylvester matrix rank function on
${\mathcal{A}}$
, unique with respect to a certain compatibility property concerning the measure
$\unicode[STIX]{x1D707}$
, namely that the rank of a characteristic function of a clopen subset
$U\subseteq X$
must equal the measure of
$U$
.
We consider skew products on
$M\times \mathbb{T}^{2}$
, where
$M$
is the two-sphere or the two-torus, which are partially hyperbolic and semi-conjugate to an Axiom A diffeomorphism. This class of dynamics includes the open sets of
$\unicode[STIX]{x1D6FA}$
-non-stable systems introduced by Abraham and Smale [Non-genericity of Ł-stability. Global Analysis (Proceedings of Symposia in Pure Mathematics, XIV (Berkeley 1968)). American Mathematical Society, Providence, RI, 1970, pp. 5–8.] and Shub [Topological Transitive Diffeomorphisms in
$T^{4}$
(Lecture Notes in Mathematics, 206). Springer, Berlin, 1971, pp. 39–40]. We present sufficient conditions, both on the skew products and the potentials, for the existence and uniqueness of equilibrium states, and discuss their statistical stability.
This paper concerns the study of the global structure of measure-preserving actions of countable groups on standard probability spaces. Weak containment is a hierarchical notion of complexity of such actions, motivated by an analogous concept in the theory of unitary representations. This concept gives rise to an associated notion of equivalence of actions, called weak equivalence, which is much coarser than the notion of isomorphism (conjugacy). It is well understood now that, in general, isomorphism is a very complex notion, a fact which manifests itself, for example, in the lack of any reasonable structure in the space of actions modulo isomorphism. On the other hand, the space of weak equivalence classes is quite well behaved. Another interesting fact that relates to the study of weak containment is that many important parameters associated with actions, such as the type, cost, and combinatorial parameters, turn out to be invariants of weak equivalence and in fact exhibit desirable monotonicity properties with respect to the pre-order of weak containment, a fact that can be useful in certain applications. There has been quite a lot of activity in this area in the last few years, and our goal in this paper is to provide a survey of this work.
A celebrated result by Bourgain and Wierdl states that ergodic averages along primes converge almost everywhere for
$L^{p}$
-functions,
$p>1$
, with a polynomial version by Wierdl and Nair. Using an anti-correlation result for the von Mangoldt function due to Green and Tao, we observe everywhere convergence of such averages for nilsystems and continuous functions.
For a non-generic, yet dense subset of
$C^{1}$
expanding Markov maps of the interval we prove the existence of uncountably many Lyapunov optimizing measures which are ergodic, fully supported and have positive entropy. These measures are equilibrium states for some Hölder continuous potentials. We also prove the existence of another non-generic dense subset for which the optimizing measure is unique and supported on a periodic orbit. A key ingredient is a new
$C^{1}$
perturbation theorem which allows us to interpolate between expanding Markov maps and the shift map on a finite number of symbols.
Kolmogorov–Sinai entropy is an invariant of measure-preserving actions of the group of integers that is central to classification theory. There are two recently developed invariants, sofic entropy and Rokhlin entropy, that generalize classical entropy to actions of countable groups. These new theories have counterintuitive properties such as factor maps that increase entropy. This survey article focusses on examples, many of which have not appeared before, that highlight the differences and similarities with classical theory.
We prove a version of the ergodic theorem for an action of an amenable group, where a Følner sequence need not be tempered. Instead, it is assumed that a function satisfies certain mixing conditions.