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It is well known that the dynamical spectrum of an ergodic measure dynamical system is related to the diffraction measure of a typical element of the system. This situation includes ergodic subshifts from symbolic dynamics as well as ergodic Delone dynamical systems, both via suitable embeddings. The connection is rather well understood when the spectrum is pure point, where the two spectral notions are essentially equivalent. In general, however, the dynamical spectrum is richer. Here, we consider (uniquely) ergodic systems of finite local complexity and establish the equivalence of the dynamical spectrum with a collection of diffraction spectra of the system and certain factors. This equivalence gives access to the dynamical spectrum via these diffraction spectra. It is particularly useful as the diffraction spectra are often simpler to determine and, in many cases, only very few of them need to be calculated.
Given a factor code
${\it\pi}$
from a shift of finite type
$X$
onto a sofic shift
$Y$
, the class degree of
${\it\pi}$
is defined to be the minimal number of transition classes over the points of
$Y$
. In this paper, we investigate the structure of transition classes and present several dynamical properties analogous to the properties of fibers of finite-to-one factor codes. As a corollary, we show that for an irreducible factor triple, there cannot be a transition between two distinct transition classes over a right transitive point, answering a question raised by Quas.
In this article we analyze totally periodic pseudo-Anosov flows in graph 3-manifolds. This means that in each Seifert fibered piece of the torus decomposition, the free homotopy class of regular fibers has a finite power which is also a finite power of the free homotopy class of a closed orbit of the flow. We show that each such flow is topologically equivalent to one of the model pseudo-Anosov flows which we previously constructed in Barbot and Fenley (Pseudo-Anosov flows in toroidal manifolds. Geom. Topol.17 (2013), 1877–1954). A model pseudo-Anosov flow is obtained by glueing standard neighborhoods of Birkhoff annuli and perhaps doing Dehn surgery on certain orbits. We also show that two model flows on the same graph manifold are isotopically equivalent (i.e. there is a isotopy of
$M$
mapping the oriented orbits of the first flow to the oriented orbits of the second flow) if and only if they have the same topological and dynamical data in the collection of standard neighborhoods of the Birkhoff annuli.
We study the dynamics of iterated function systems generated by a pair of circle diffeomorphisms close to rotations in the
$C^{1+\text{bv}}$
-topology. We characterize the obstruction to minimality and describe the limit set. In particular, there are no invariant minimal Cantor sets, which can be seen as a Denjoy/Duminy type theorem for iterated systems on the circle.
In this paper we consider
${C}^{1+ \epsilon } $
area-preserving diffeomorphisms of the torus
$f$
, either homotopic to the identity or to Dehn twists. We suppose that
$f$
has a lift
$\widetilde {f} $
to the plane such that its rotation set has interior and prove, among other things, that if zero is an interior point of the rotation set, then there exists a hyperbolic
$\widetilde {f} $
-periodic point
$\widetilde {Q} \in { \mathbb{R} }^{2} $
such that
${W}^{u} (\widetilde {Q} )$
intersects
${W}^{s} (\widetilde {Q} + (a, b))$
for all integers
$(a, b)$
, which implies that
$ \overline{{W}^{u} (\widetilde {Q} )} $
is invariant under integer translations. Moreover,
$ \overline{{W}^{u} (\widetilde {Q} )} = \overline{{W}^{s} (\widetilde {Q} )} $
and
$\widetilde {f} $
restricted to
$ \overline{{W}^{u} (\widetilde {Q} )} $
is invariant and topologically mixing. Each connected component of the complement of
$ \overline{{W}^{u} (\widetilde {Q} )} $
is a disk with diameter uniformly bounded from above. If
$f$
is transitive, then
$ \overline{{W}^{u} (\widetilde {Q} )} = { \mathbb{R} }^{2} $
and
$\widetilde {f} $
is topologically mixing in the whole plane.
We study limit sets of stable cellular automata from a symbolic dynamics point of view, where they are a special case of sofic shifts admitting a steady epimorphism. We prove that there exists a right-closing almost-everywhere steady factor map from one irreducible sofic shift onto another one if and only if there exists such a map from the domain onto the minimal right-resolving cover of the image. We define right-continuing almost-everywhere steady maps, and prove that there exists such a steady map between two sofic shifts if and only if there exists a factor map from the domain onto the minimal right-resolving cover of the image. To translate this into terms of cellular automata, a sofic shift can be the limit set of a stable cellular automaton with a right-closing almost-everywhere dynamics onto its limit set if and only if it is the factor of a full shift and there exists a right-closing almost-everywhere factor map from the sofic shift onto its minimal right-resolving cover. A sofic shift can be the limit set of a stable cellular automaton reaching its limit set with a right-continuing almost-everywhere factor map if and only if it is the factor of a full shift and there exists a factor map from the sofic shift onto its minimal right-resolving cover. Finally, as a consequence of the previous results, we provide a characterization of the almost of finite type shifts (AFT) in terms of a property of steady maps that have them as range.
Let
$ \mathbb{B} $
be a
$p$
-uniformly convex Banach space, with
$p\geq 2$
. Let
$T$
be a linear operator on
$ \mathbb{B} $
, and let
${A}_{n} x$
denote the ergodic average
$(1/ n){\mathop{\sum }\nolimits}_{i\lt n} {T}^{n} x$
. We prove the following variational inequality in the case where
$T$
is power bounded from above and below: for any increasing sequence
$\mathop{({t}_{k} )}\nolimits_{k\in \mathbb{N} } $
of natural numbers we have
${\mathop{\sum }\nolimits}_{k} \mathop{\Vert {A}_{{t}_{k+ 1} } x- {A}_{{t}_{k} } x\Vert }\nolimits ^{p} \leq C\mathop{\Vert x\Vert }\nolimits ^{p} $
, where the constant
$C$
depends only on
$p$
and the modulus of uniform convexity. For
$T$
a non-expansive operator, we obtain a weaker bound on the number of
$\varepsilon $
-fluctuations in the sequence. We clarify the relationship between bounds on the number of
$\varepsilon $
-fluctuations in a sequence and bounds on the rate of metastability, and provide lower bounds on the rate of metastability that show that our main result is sharp.
In this paper we characterize
$\omega $
-limit sets of dendritic Julia sets for quadratic maps. We use Baldwin’s symbolic representation of these spaces as a non-Hausdorff itinerary space and prove that quadratic maps with dendritic Julia sets have shadowing, and also that for all such maps, a closed invariant set is an
$\omega $
-limit set of a point if, and only if, it is internally chain transitive.
We solve several problems on frequently hypercyclic operators. Firstly, we characterize frequently hypercyclic weighted shifts on
${\ell }^{p} ( \mathbb{Z} )$
,
$p\geq 1$
. Our method uses properties of the difference set of a set with positive upper density. Secondly, we show that there exists an operator which is
$ \mathcal{U} $
-frequently hypercyclic but not frequently hypercyclic, and that there exists an operator which is frequently hypercyclic but not distributionally chaotic. These (surprising) counterexamples are given by weighted shifts on
${c}_{0} $
. The construction of these shifts lies on the construction of sets of positive integers whose difference sets have very specific properties.
We obtain new characterizations of Li–Yorke chaos for linear operators on Banach and Fréchet spaces. We also offer conditions under which an operator admits a dense set or linear manifold of irregular vectors. Some of our general results are applied to composition operators and adjoint multipliers on spaces of holomorphic functions.
We describe the behaviour at infinity of the bifurcation current in the moduli space of quadratic rational maps. To this purpose, we extend it to some closed, positive
$(1,1)$
-current on a two-dimensional complex projective space and then compute the Lelong numbers and the self-intersection of the extended current.
Consider the space of sequences of
$k$
letters ordered lexicographically. We study the set
${\mathcal{M}}(\boldsymbol{{\it\alpha}})$
of all maximal sequences for which the asymptotic proportions
$\boldsymbol{{\it\alpha}}$
of the letters are prescribed, where a sequence is said to be maximal if it is at least as great as all of its tails. The infimum of
${\mathcal{M}}(\boldsymbol{{\it\alpha}})$
is called the
$\boldsymbol{{\it\alpha}}$
-infimax sequence, or the
$\boldsymbol{{\it\alpha}}$
-minimax sequence if the infimum is a minimum. We give an algorithm which yields all infimax sequences, and show that the infimax is not a minimax if and only if it is the
$\boldsymbol{{\it\alpha}}$
-infimax for every
$\boldsymbol{{\it\alpha}}$
in a simplex of dimension 1 or greater. These results have applications to the theory of rotation sets of beta-shifts and torus homeomorphisms.
We use an Ulam-type discretization scheme to provide pointwise approximations for invariant densities of interval maps with a neutral fixed point. We prove that the approximate invariant density converges pointwise to the true density at a rate
${C}^{\ast } \cdot (\ln m)/ m$
, where
${C}^{\ast } $
is a computable fixed constant and
${m}^{- 1} $
is the mesh size of the discretization.
We extend the classical notion of block structure for periodic orbits of interval maps to the setting of tree maps and study the algebraic properties of the Markov matrix of a periodic tree pattern having a block structure. We also prove a formula which relates the topological entropy of a pattern having a block structure with that of the underlying periodic pattern obtained by collapsing each block to a point, and characterize the structure of the zero entropy patterns in terms of block structures. Finally, we prove that an
$n$
-periodic pattern has zero (positive) entropy if and only if all
$n$
-periodic patterns obtained by considering the
$k\mathrm{th} $
iterate of the map on the invariant set have zero (respectively, positive) entropy, for each
$k$
relatively prime to
$n$
.
We consider unimodular random rooted trees (URTs) and invariant forests in Cayley graphs. We show that URTs of bounded degree are the same as the law of the component of the root in an invariant percolation on a regular tree. We use this to give a new proof that URTs are sofic, a result of Elek. We show that ends of invariant forests in the hyperbolic plane converge to ideal boundary points. We also note that uniform integrability of the degree distribution of a family of finite graphs implies tightness of that family for local convergence, also known as random weak convergence.
We show that, under the assumption of chain transitivity, the shadowing property is equivalent to the thick shadowing property. We also show that, if
${\mathcal{F}}$
is a family with the Ramsey property, then an arbitrary sequence of points in a chain transitive space can be
${\it\varepsilon}$
-shadowed (for any
${\it\varepsilon}$
) on a set in
${\mathcal{F}}$
.
Let
$G$
be a Hausdorff, étale groupoid that is minimal and topologically principal. We show that
$C_{r}^{\ast }(G)$
is purely infinite simple if and only if all the non-zero positive elements of
$C_{0}(G^{(0)})$
are infinite in
$C_{r}^{\ast }(G)$
. If
$G$
is a Hausdorff, ample groupoid, then we show that
$C_{r}^{\ast }(G)$
is purely infinite simple if and only if every non-zero projection in
$C_{0}(G^{(0)})$
is infinite in
$C_{r}^{\ast }(G)$
. We then show how this result applies to
$k$
-graph
$C^{\ast }$
-algebras. Finally, we investigate strongly purely infinite groupoid
$C^{\ast }$
-algebras.
In this paper we consider quadratic polynomials on the complex plane
${f}_{c} (z)= {z}^{2} + c$
and their associated Julia sets,
${J}_{c} $
. Specifically, we consider the case that the kneading sequence is periodic and not an
$n$
-tupling. In this case
${J}_{c} $
contains subsets that are homeomorphic to the unit circle, usually infinitely many disjoint such subsets. We prove that
${f}_{c} : {J}_{c} \rightarrow {J}_{c} $
has shadowing, and we classify all
$\omega $
-limit sets for these maps by showing that a closed set
$R\subseteq {J}_{c} $
is internally chain transitive if, and only if, there is some
$z\in {J}_{c} $
with
$\omega (z)= R$
.
Let
$T_{1}$
and
$T_{2}$
be two commuting probability measure-preserving actions of a countable amenable group such that the group spanned by these actions acts ergodically. We show that
${\it\mu}(A\cap T_{1}^{g}A\cap T_{1}^{g}T_{2}^{g}A)>{\it\mu}(A)^{4}-{\it\epsilon}$
on a syndetic set for any measurable set
$A$
and any
${\it\epsilon}>0$
. The proof uses the concept of a sated system, introduced by Austin.
We study invariant Fatou components for holomorphic endomorphisms in
$\mathbb{P}^{2}$
. In the recurrent case these components were classified by Fornæss and Sibony [Classification of recurrent domains for some holomorphic maps. Math. Ann.301(4) (1995), 813–820]. Ueda [Holomorphic maps on projective spaces and continuations of Fatou maps. Michigan Math J.56(1) (2008), 145–153] completed this classification by proving that it is not possible for the limit set to be a punctured disk. Recently Lyubich and Peters [Classification of invariant Fatou components for dissipative Hénon maps. Preprint] classified non-recurrent invariant Fatou components, under the additional hypothesis that the limit set is unique. Again all possibilities in this classification were known to occur, except for the punctured disk. Here we show that the punctured disk can indeed occur as the limit set of a non-recurrent Fatou component. We provide many additional examples of holomorphic and polynomial endomorphisms of
$\mathbb{C}^{2}$
with non-recurrent Fatou components on which the orbits converge to the regular part of arbitrary analytic sets.