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We prove that, although the map is singular, its square preserves the Lebesgue measure and is strongly mixing, thus ergodic, with respect to it. We discuss the extension of the results to more general erasing maps.
We examine multiple ergodic averages of commuting transformations with polynomial iterates in which the polynomials may be pairwise dependent. In particular, we show that such averages are controlled by the Gowers–Host–Kra seminorms whenever the system satisfies some mild ergodicity assumptions. Combining this result with the general criteria for joint ergodicity established in our earlier work, we determine a necessary and sufficient condition under which such averages are jointly ergodic, in the sense that they converge in the mean to the product of integrals, or weakly jointly ergodic, in that they converge to the product of conditional expectations. As a corollary, we deduce a special case of a conjecture by Donoso, Koutsogiannis, and Sun in a stronger form.
We use Gaussian measure-preserving systems to prove the existence and genericity of Lebesgue measure-preserving transformations
$T:[0,1]\rightarrow [0,1]$
which exhibit both mixing and rigidity behavior along families of asymptotically linearly independent sequences. Let
$\unicode{x3bb} _1,\ldots ,\unicode{x3bb} _N\in [0,1]$
and let
$\phi _1,\ldots ,\phi _N:\mathbb N\rightarrow \mathbb Z$
be asymptotically linearly independent (that is, for any
$(a_1,\ldots ,a_N)\in \mathbb Z^N\setminus \{\vec 0\}$
,
$\lim _{k\rightarrow \infty }|\sum _{j=1}^Na_j\phi _j(k)|=\infty $
). Then the class of invertible Lebesgue measure-preserving transformations
$T:[0,1]\rightarrow [0,1]$
for which there exists a sequence
$(n_k)_{k\in \mathbb {N}}$
in
$\mathbb {N}$
with for any measurable
$A,B\subseteq [0,1]$
and any
$j\in \{1,\ldots ,N\}$
, is generic. This result is a refinement of a result due to Stëpin (Theorem 2 in [Spectral properties of generic dynamical systems. Math. USSR-Izv.29(1) (1987), 159–192]) and a generalization of a result due to Bergelson, Kasjan, and Lemańczyk (Corollary F in [Polynomial actions of unitary operators and idempotent ultrafilters. Preprint, 2014, arXiv:1401.7869]).
A fundamental question in the field of cohomology of dynamical systems is to determine when there are solutions to the coboundary equation:
$$ \begin{align*} f = g - g \circ T. \end{align*} $$
In many cases, T is given to be an ergodic invertible measure-preserving transformation on a standard probability space
$(X, {\mathcal B}, \mu )$
and is contained in
$L^p$
for
$p \geq 0$
. We extend previous results by showing for any measurable f that is non-zero on a set of positive measure, the class of measure-preserving T with a measurable solution g is meager (including the case where
$\int _X f\,d\mu = 0$
). From this fact, a natural question arises: given f, does there always exist a solution pair T and g? In regards to this question, our main results are as follows. Given measurable f, there exist an ergodic invertible measure-preserving transformation T and measurable function g such that
$f(x) = g(x) - g(Tx)$
for almost every (a.e.)
$x\in X$
, if and only if
$\int _{f> 0} f\,d\mu = - \int _{f < 0} f\,d\mu $
(whether finite or
$\infty $
). Given mean-zero
$f \in L^p(\mu )$
for
$p \geq 1$
, there exist an ergodic invertible measure-preserving T and
$g \in L^{p-1}(\mu )$
such that
$f(x) = g(x) - g( Tx )$
for a.e.
$x \in X$
. In some sense, the previous existence result is the best possible. For
$p \geq 1$
, there exists a dense
$G_{\delta }$
set of mean-zero
$f \in L^p(\mu )$
such that for any ergodic invertible measure-preserving T and any measurable g such that
$f(x) = g(x) - g(Tx)$
almost everywhere, then
$g \notin L^q(\mu )$
for
$q> p - 1$
. Finally, it is shown that we cannot expect finite moments for solutions g, when
$f \in L^1(\mu )$
. In particular, given any such that
$\lim _{x\to \infty } \phi (x) = \infty $
, there exist mean-zero
$f \in L^1(\mu )$
such that for any solutions T and g, the transfer function g satisfies:
We establish the mean convergence for multiple ergodic averages with iterates given by distinct fractional powers of primes and related multiple recurrence results. A consequence of our main result is that every set of integers with positive upper density contains patterns of the form
$\{m,m+[p_n^a], m+[p_n^b]\}$
, where
$a,b$
are positive nonintegers and
$p_n$
denotes the nth prime, a property that fails if a or b is a natural number. Our approach is based on a recent criterion for joint ergodicity of collections of sequences, and the bulk of the proof is devoted to obtaining good seminorm estimates for the related multiple ergodic averages. The input needed from number theory are upper bounds for the number of prime k-tuples that follow from elementary sieve theory estimates and equidistribution results of fractional powers of primes in the circle.
The f-invariant is an isomorphism invariant of free-group measure-preserving actions introduced by Lewis Bowen, who first used it to show that two finite-entropy Bernoulli shifts over a finitely generated free group can be isomorphic only if their base measures have the same Shannon entropy. Bowen also showed that the f-invariant is a variant of sofic entropy; in particular, it is the exponential growth rate of the expected number of good models over a uniform random homomorphism. In this paper we present an analogous formula for the relative f-invariant and use it to prove a formula for the exponential growth rate of the expected number of good models over a random sofic approximation which is a type of stochastic block model.
We generalize the greedy and lazy
$\beta $
-transformations for a real base
$\beta $
to the setting of alternate bases
${\boldsymbol {\beta }}=(\beta _0,\ldots ,\beta _{p-1})$
, which were recently introduced by the first and second authors as a particular case of Cantor bases. As in the real base case, these new transformations, denoted
$T_{{\boldsymbol {\beta }}}$
and
$L_{{\boldsymbol {\beta }}}$
respectively, can be iterated in order to generate the digits of the greedy and lazy
${\boldsymbol {\beta }}$
-expansions of real numbers. The aim of this paper is to describe the measure-theoretical dynamical behaviors of
$T_{{\boldsymbol {\beta }}}$
and
$L_{{\boldsymbol {\beta }}}$
. We first prove the existence of a unique absolutely continuous (with respect to an extended Lebesgue measure, called the p-Lebesgue measure)
$T_{{\boldsymbol {\beta }}}$
-invariant measure. We then show that this unique measure is in fact equivalent to the p-Lebesgue measure and that the corresponding dynamical system is ergodic and has entropy
$({1}/{p})\log (\beta _{p-1}\cdots \beta _0)$
. We give an explicit expression of the density function of this invariant measure and compute the frequencies of letters in the greedy
${\boldsymbol {\beta }}$
-expansions. The dynamical properties of
$L_{{\boldsymbol {\beta }}}$
are obtained by showing that the lazy dynamical system is isomorphic to the greedy one. We also provide an isomorphism with a suitable extension of the
$\beta $
-shift. Finally, we show that the
${\boldsymbol {\beta }}$
-expansions can be seen as
$(\beta _{p-1}\cdots \beta _0)$
-representations over general digit sets and we compare both frameworks.
In this paper I argue for an association between impurity and explanatory power in contemporary mathematics. This proposal is defended against the ancient and influential idea that purity and explanation go hand-in-hand (Aristotle, Bolzano) and recent suggestions that purity/impurity ascriptions and explanatory power are more or less distinct (Section 1). This is done by analyzing a central and deep result of additive number theory, Szemerédi’s theorem, and various of its proofs (Section 2). In particular, I focus upon the radically impure (ergodic) proof due to Furstenberg (Section 3). Furstenberg’s ergodic proof is striking because it utilizes intuitively foreign and infinitary resources to prove a finitary combinatorial result and does so in a perspicuous fashion. I claim that Furstenberg’s proof is explanatory in light of its clear expression of a crucial structural result, which provides the “reason why” Szemerédi’s theorem is true. This is, however, rather surprising: how can such intuitively different conceptual resources “get a grip on” the theorem to be proved? I account for this phenomenon by articulating a new construal of the content of a mathematical statement, which I call structural content (Section 4). I argue that the availability of structural content saves intuitive epistemic distinctions made in mathematical practice and simultaneously explicates the intervention of surprising and explanatorily rich conceptual resources. Structural content also disarms general arguments for thinking that impurity and explanatory power might come apart. Finally, I sketch a proposal that, once structural content is in hand, impure resources lead to explanatory proofs via suitably understood varieties of simplification and unification (Section 5).
Let
$\mathcal {P}$
be an (unbounded) countable multiset of primes (that is, every prime may appear multiple times) and let
$G=\bigoplus _{p\in \mathcal {P}}\mathbb {F}_p$
. We develop a Host–Kra structure theory for the universal characteristic factors of an ergodic G-system. More specifically, we generalize the main results of Bergelson, Tao and Ziegler [An inverse theorem for the uniformity seminorms associated with the action of
$\mathbb {F}_p^\infty $
. Geom. Funct. Anal.19(6) (2010), 1539–1596], who studied these factors in the special case
$\mathcal {P}=\{p,p,p,\ldots \}$
for some fixed prime p. As an application we deduce a Khintchine-type recurrence theorem in the flavor of Bergelson, Tao and Ziegler [Multiple recurrence and convergence results associated to
$F_p^\omega $
-actions. J. Anal. Math.127 (2015), 329–378] and Bergelson, Host and Kra [Multiple recurrence and nilsequences. Invent. Math.160(2) (2005), 261–303, with an appendix by I. Ruzsa].
Packing topological entropy is a dynamical analogy of the packing dimension, which can be viewed as a counterpart of Bowen topological entropy. In the present paper we give a systematic study of the packing topological entropy for a continuous G-action dynamical system
$(X,G)$
, where X is a compact metric space and G is a countable infinite discrete amenable group. We first prove a variational principle for amenable packing topological entropy: for any Borel subset Z of X, the packing topological entropy of Z equals the supremum of upper local entropy over all Borel probability measures for which the subset Z has full measure. Then we obtain an entropy inequality concerning amenable packing entropy. Finally, we show that the packing topological entropy of the set of generic points for any invariant Borel probability measure
$\mu $
coincides with the metric entropy if either
$\mu $
is ergodic or the system satisfies a kind of specification property.
Let X be a compact, geodesically complete, locally CAT(0) space such that the universal cover admits a rank-one axis. Assume X is not homothetic to a metric graph with integer edge lengths. Let
$P_t$
be the number of parallel classes of oriented closed geodesics of length at most t; then
$\lim \nolimits _{t \to \infty } P_t / ({e^{ht}}/{ht}) = 1$
, where h is the entropy of the geodesic flow on the space
$GX$
of parametrized unit-speed geodesics in X.
We show that self-similar measures on
$\mathbb R^d$
satisfying the weak separation condition are uniformly scaling. Our approach combines elementary ergodic theory with geometric analysis of the structure given by the weak separation condition.
In this paper we prove a local exponential synchronization for Markovian random iterations of homeomorphisms of the circle
$S^{1}$
, providing a new result on stochastic circle dynamics even for
$C^1$
-diffeomorphisms. This result is obtained by combining an invariance principle for stationary random iterations of homeomorphisms of the circle with a Krylov–Bogolyubov-type result for homogeneous Markov chains.
Entropy dimension is an entropy-type quantity which takes values in
$[0,1]$
and classifies different levels of intermediate growth rate of complexity for dynamical systems. In this paper, we consider the complexity of skew products of irrational rotations with Bernoulli systems, which can be viewed as deterministic walks in random sceneries, and show that this class of models can have any given entropy dimension by choosing suitable rotations for the base system.
Given a closed, orientable, compact surface S of constant negative curvature and genus
$g \geq 2$
, we study the measure-theoretic entropy of the Bowen–Series boundary map with respect to its smooth invariant measure. We obtain an explicit formula for the entropy that only depends on the perimeter of the
$(8g-4)$
-sided fundamental polygon of the surface S and its genus. Using this, we analyze how the entropy changes in the Teichmüller space of S and prove the following flexibility result: the measure-theoretic entropy takes all values between 0 and a maximum that is achieved on the surface that admits a regular
$(8g-4)$
-sided fundamental polygon. We also compare the measure-theoretic entropy to the topological entropy of these maps and show that the smooth invariant measure is not a measure of maximal entropy.
We study the dynamical Borel–Cantelli lemma for recurrence sets in a measure-preserving dynamical system
$(X, \mu , T)$
with a compatible metric d. We prove that under some regularity conditions, the
$\mu $
-measure of the following set
obeys a zero–full law according to the convergence or divergence of a certain series, where
$\psi :\mathbb {N}\to \mathbb {R}^+$
. The applications of our main theorem include the Gauss map,
$\beta $
-transformation and homogeneous self-similar sets.
We show that every countable group with infinite finite conjugacy (FC)-center has the Schmidt property, that is, admits a free, ergodic, measure-preserving action on a standard probability space such that the full group of the associated orbit equivalence relation contains a non-trivial central sequence. As a consequence, every countable, inner amenable group with property (T) has the Schmidt property.
We consider a strictly substochastic matrix or a stochastic matrix with absorbing states. By using quasi-stationary distributions we show that there is an associated canonical Markov chain that is built from the resurrected chain, the absorbing states, and the hitting times, together with a random walk on the absorbing states, which is necessary for achieving time stationarity. Based upon the 2-stringing representation of the resurrected chain, we supply a stationary representation of the killed and the absorbed chains. The entropies of these representations have a clear meaning when one identifies the probability measure of natural factors. The balance between the entropies of these representations and the entropy of the canonical chain serves to check the correctness of the whole construction.
Let f be a
$C^2$
diffeomorphism on a compact manifold. Ledrappier and Young introduced entropies along unstable foliations for an ergodic measure
$\mu $
. We relate those entropies to covering numbers in order to give a new upper bound on the metric entropy of
$\mu $
in terms of Lyapunov exponents and topological entropy or volume growth of sub-manifolds. We also discuss extensions to the
$C^{1+\alpha },\,\alpha>0$
, case.
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.