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Given a subshift
$\Sigma $
of finite type and a finite set S of finite words, let
$\Sigma \langle S\rangle $
denote the subshift of
$\Sigma $
that avoids S. We establish a general criterion under which we can bound the entropy perturbation
$h(\Sigma ) - h(\Sigma \langle S\rangle )$
from above. As an application, we prove that this entropy difference tends to zero with a sequence of such sets
$S_1, S_2,\ldots $
under various assumptions on the
$S_i$
.
Let G be a countably infinite discrete amenable group. It should be noted that a G-system
$(X,G)$
naturally induces a G-system
$(\mathcal {M}(X),G)$
, where
$\mathcal {M}(X)$
denotes the space of Borel probability measures on the compact metric space X endowed with the weak*-topology. A factor map
$\pi : (X,G)\to (Y,G)$
between two G-systems induces a factor map
$\widetilde {\pi }:(\mathcal {M}(X),G)\to (\mathcal {M}(Y),G)$
. It turns out that
$\widetilde {\pi }$
is open if and only if
$\pi $
is open. When Y is fully supported, it is shown that
$\pi $
has relative uniformly positive entropy if and only if
$\widetilde {\pi }$
has relative uniformly positive entropy.
We provide an explicit
$\mathcal {S}$
-adic representation of rank-one subshifts with bounded spacers and call the subshifts obtained in this way ‘minimal Ferenczi subshifts’. We aim to show that this approach is very convenient to study the dynamical behavior of rank-one systems. For instance, we compute their topological rank, the strong and the weak orbit equivalence class. We observe that they have an induced system that is a Toeplitz subshift having discrete spectrum. We also characterize continuous and non-continuous eigenvalues of minimal Ferenczi subshifts.
Using the idea of local entropy theory, we characterize the sequence entropy tuple via mean forms of the sensitive tuple in both topological and measure-theoretical senses. For the measure-theoretical sense, we show that for an ergodic measure-preserving system, the
$\mu $
-sequence entropy tuple, the
$\mu $
-mean sensitive tuple, and the
$\mu $
-sensitive in the mean tuple coincide, and give an example to show that the ergodicity condition is necessary. For the topological sense, we show that for a certain class of minimal systems, the mean sensitive tuple is the sequence entropy tuple.
In this paper, we study almost proximal extensions of minimal flows. Let
$\pi : (X,T)\rightarrow (Y,T)$
be an extension of minimal flows. Then
$\pi $
is called an almost proximal extension if there is some
$N\in {\mathbb N}$
such that the cardinality of any almost periodic subset in each fiber is not greater than N. When
$N=1$
,
$\pi $
is proximal. We will give the structure of
$\pi $
and give a dichotomy theorem: any almost proximal extension of minimal flows is either almost finite to one, or almost all fibers contain an uncountable strongly scrambled subset. Using the category method, Glasner and Weiss showed the existence of proximal but not almost one-to-one extensions [On the construction of minimal skew products. Israel J. Math.34 (1979), 321–336]. In this paper, we will give explicit such examples, and also examples of almost proximal but not almost finite to one extensions.
We investigate several questions related to the notion of recognizable morphism. The main result is a new proof of Mossé’s theorem and actually of a generalization to a more general class of morphisms due to Berthé et al [Recognizability for sequences of morphisms. Ergod. Th. & Dynam. Sys.39(11) (2019), 2896–2931]. We actually prove the result of Berthé et al for the most general class of morphisms, including ones with erasable letters. Our result is derived from a result concerning elementary morphisms for which we also provide a new proof. We also prove some new results which allow us to formulate the property of recognizability in terms of finite automata. We use this characterization to show that for an injective morphism, the property of being recognizable on the full shift for aperiodic points is decidable.
We give a characterization of inter-model sets with Euclidean internal space. This characterization is similar to previous results for general inter-model sets obtained independently by Baake, Lenz and Moody, and Aujogue. The new ingredients are two additional conditions. The first condition is on the rank of the abelian group generated by the set of internal differences. The second condition is on a flow on a torus defined via the address map introduced by Lagarias. This flow plays the role of the maximal equicontinuous factor in the previous characterizations.
In this paper, we address the problem of computing the topological entropy of a map
$\psi : G \to G$
, where G is a Lie group, given by some power
$\psi (g) = g^k$
, with k a positive integer. When G is abelian,
$\psi $
is an endomorphism and its topological entropy is given by
$h(\psi ) = \dim (T(G)) \log (k)$
, where
$T(G)$
is the maximal torus of G, as shown by Patrão [The topological entropy of endomorphisms of Lie groups. Israel J. Math.234 (2019), 55–80]. However, when G is not abelian,
$\psi $
is no longer an endomorphism and these previous results cannot be used. Still,
$\psi $
has some interesting symmetries, for example, it commutes with the conjugations of G. In this paper, the structure theory of Lie groups is used to show that
$h(\psi ) = \dim (T)\log (k)$
, where T is a maximal torus of G, generalizing the formula in the abelian case. In particular, the topological entropy of powers on compact Lie groups with discrete center is always positive, in contrast to what happens to endomorphisms of such groups, which always have null entropy.
We prove an explicit characterization of the points in Thurston’s Master Teapot, which can be implemented algorithmically to test whether a point in $\mathbb {C}\times \mathbb {R}$ belongs to the complement of the Master Teapot. As an application, we show that the intersection of the Master Teapot with the unit cylinder is not symmetrical under reflection through the plane that is the product of the imaginary axis of $\mathbb {C}$ and $\mathbb {R}$.
We study recurrence in the real quadratic family and give a sufficient condition on the recurrence rate
$(\delta _n)$
of the critical orbit such that, for almost every non-regular parameter a, the set of n such that
$\vert F^n(0;a) \vert < \delta _n$
is infinite. In particular, when
$\delta _n = n^{-1}$
, this extends an earlier result by Avila and Moreira [Statistical properties of unimodal maps: the quadratic family. Ann. of Math. (2)161(2) (2005), 831–881].
Let X be a compact metric space and let $f: X\!\rightarrow \! X$ be a homeomorphism on X. We show that if f is both pointwise recurrent and expansive, then the dynamical system $(X, f)$ is topologically conjugate to a subshift of some symbolic system. Moreover, if f is pointwise positively recurrent, then the subshift is semisimple; a counterexample is given to show the necessity of positive recurrence to ensure the semisimplicity.
We show that for a Salem number
$\beta $
of degree d, there exists a positive constant
$c(d)$
where
$\beta ^m$
is a Parry number for integers m of natural density
$\ge c(d)$
. Further, we show
$c(6)>1/2$
and discuss a relation to the discretized rotation in dimension
$4$
.
We prove that any continuous function can be locally approximated at a fixed point
$x_{0}$
by an uncountable family resistant to disruptions by the family of continuous functions for which
$x_{0}$
is a fixed point. In that context, we also consider the property of quasicontinuity.
Let $k\geq 2$ and $(X_{i}, \mathcal {T}_{i}), i=1,\ldots ,k$, be $\mathbb {Z}^{d}$-actions topological dynamical systems with $\mathcal {T}_i:=\{T_i^{\textbf {g}}:X_i{\rightarrow } X_i\}_{\textbf {g}\in \mathbb {Z}^{d}}$, where $d\in \mathbb {N}$ and $f\in C(X_{1})$. Assume that for each $1\leq i\leq k-1$, $(X_{i+1}, \mathcal {T}_{i+1})$ is a factor of $(X_{i}, \mathcal {T}_{i})$. In this paper, we introduce the weighted topological pressure $P^{\textbf {a}}(\mathcal {T}_{1},f)$ and weighted measure-theoretic entropy $h_{\mu }^{\textbf {a}}(\mathcal {T}_{1})$ for $\mathbb {Z}^{d}$-actions, and establish a weighted variational principle as
This result not only generalizes some well-known variational principles about topological pressure for compact or non-compact sets, but also improves the variational principle for weighted topological pressure in [16] from $\mathbb {Z}_{+}$-action topological dynamical systems to $\mathbb {Z}^{d}$-actions topological dynamical systems.
A
$D_{\infty }$
-topological Markov chain is a topological Markov chain provided with an action of the infinite dihedral group
$D_{\infty }$
. It is defined by two zero-one square matrices A and J satisfying
$AJ=JA^{\textsf {T}}$
and
$J^2=I$
. A flip signature is obtained from symmetric bilinear forms with respect to J on the eventual kernel of A. We modify Williams’ decomposition theorem to prove the flip signature is a
$D_{\infty }$
-conjugacy invariant. We introduce natural
$D_{\infty }$
-actions on Ashley’s eight-by-eight and the full two-shift. The flip signatures show that Ashley’s eight-by-eight and the full two-shift equipped with the natural
$D_{\infty }$
-actions are not
$D_{\infty }$
-conjugate. We also discuss the notion of
$D_{\infty }$
-shift equivalence and the Lind zeta function.
We prove that for any non-degenerate dendrite D, there exist topologically mixing maps
$F : D \to D$
and
$f : [0, 1] \to [0, 1]$
such that the natural extensions (as known as shift homeomorphisms)
$\sigma _F$
and
$\sigma _f$
are conjugate, and consequently the corresponding inverse limits are homeomorphic. Moreover, the map f does not depend on the dendrite D and can be selected so that the inverse limit
$\underleftarrow {\lim } (D,F)$
is homeomorphic to the pseudo-arc. The result extends to any finite number of dendrites. Our work is motivated by, but independent of, the recent result of the first and third author on conjugation of Lozi and Hénon maps to natural extensions of dendrite maps.
Given a minimal action
$\alpha $
of a countable group on the Cantor set, we show that the alternating full group
$\mathsf {A}(\alpha )$
is non-amenable if and only if the topological full group
$\mathsf {F}(\alpha )$
is
$C^*$
-simple. This implies, for instance, that the Elek–Monod example of non-amenable topological full group coming from a Cantor minimal
$\mathbb {Z}^2$
-system is
$C^*$
-simple.
Sarnak’s Möbius disjointness conjecture asserts that for any zero entropy dynamical system
$(X,T)$
,
$({1}/{N})\! \sum _{n=1}^{N}\! f(T^{n} x) \mu (n)= o(1)$
for every
$f\in \mathcal {C}(X)$
and every
$x\in X$
. We construct examples showing that this
$o(1)$
can go to zero arbitrarily slowly. In fact, our methods yield a more general result, where in lieu of
$\mu (n)$
, one can put any bounded sequence
$a_{n}$
such that the Cesàro mean of the corresponding sequence of absolute values does not tend to zero. Moreover, in our construction, the choice of x depends on the sequence
$a_{n}$
but
$(X,T)$
does not.
In this work, we study the entropies of subsystems of shifts of finite type (SFTs) and sofic shifts on countable amenable groups. We prove that for any countable amenable group G, if X is a G-SFT with positive topological entropy
$h(X)> 0$
, then the entropies of the SFT subsystems of X are dense in the interval
$[0, h(X)]$
. In fact, we prove a ‘relative’ version of the same result: if X is a G-SFT and
$Y \subset X$
is a subshift such that
$h(Y) < h(X)$
, then the entropies of the SFTs Z for which
$Y \subset Z \subset X$
are dense in
$[h(Y), h(X)]$
. We also establish analogous results for sofic G-shifts.
We investigate dynamical systems consisting of a locally compact Hausdorff space equipped with a partially defined local homeomorphism. Important examples of such systems include self-covering maps, one-sided shifts of finite type and, more generally, the boundary-path spaces of directed and topological graphs. We characterize the topological conjugacy of these systems in terms of isomorphisms of their associated groupoids and C*-algebras. This significantly generalizes recent work of Matsumoto and of the second- and third-named authors.