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In the pioneering work by Dimitrov–Haiden–Katzarkov–Kontsevich, they introduced various categorical analogies from the classical theory of dynamical systems. In particular, they defined the entropy of an endofunctor on a triangulated category with a split generator. In the connection between the categorical theory and the classical theory, a stability condition on a triangulated category plays the role of a measured foliation so that one can measure the “volume” of objects, called the mass, via the stability condition. The aim of this paper is to establish fundamental properties of the growth rate of mass of objects under the mapping by the endofunctor and to clarify the relationship between it and the entropy. We also show that they coincide under a certain condition.
In this work, we treat subshifts, defined in terms of an alphabet $\mathcal {A}$ and (usually infinite) forbidden list $\mathcal {F}$, where the number of n-letter words in $\mathcal {F}$ has ‘slow growth rate’ in n. We show that such subshifts are well behaved in several ways; for instance, they are boundedly supermultiplicative in the sense of Baker and Ghenciu [Dynamical properties of S-gap shifts and other shift spaces. J. Math. Anal. Appl.430(2) (2015), 633–647] and they have unique measures of maximal entropy with the K-property and which satisfy Gibbs bounds on large (measure-theoretically) sets. The main tool in our proofs is a more general result, which states that bounded supermultiplicativity and a sort of measure-theoretic specification property together imply uniqueness of the measure of maximum entropy and our Gibbs bounds. We also show that some well-known classes of subshifts can be treated by our results, including the symbolic codings of $x \mapsto \alpha + \beta x$ (the so-called $\alpha $-$\beta $ shifts of Hofbauer [Maximal measures for simple piecewise monotonic transformations. Z. Wahrsch. verw. Geb.52(3) (1980), 289–300]) and the bounded density subshifts of Stanley [Bounded density shifts. Ergod. Th. & Dynam. Sys.33(6) (2013), 1891–1928].
Let
$(X,T)$
be a topological dynamical system consisting of a compact metric space X and a continuous surjective map
$T : X \to X$
. By using local entropy theory, we prove that
$(X,T)$
has uniformly positive entropy if and only if so does the induced system
$({\mathcal {M}}(X),\widetilde {T})$
on the space of Borel probability measures endowed with the weak* topology. This result can be seen as a version for the notion of uniformly positive entropy of the corresponding result for topological entropy due to Glasner and Weiss.
We prove that for $C^0$-generic homeomorphisms, acting on a compact smooth boundaryless manifold with dimension greater than one, the upper metric mean dimension with respect to the smooth metric coincides with the dimension of the manifold. As an application, we show that the upper box dimension of the set of periodic points of a $C^0$-generic homeomorphism is equal to the dimension of the manifold. In the case of continuous interval maps, we prove that each level set for the metric mean dimension with respect to the Euclidean distance is $C^0$-dense in the space of continuous endomorphisms of $[0,1]$ with the uniform topology. Moreover, the maximum value is attained at a $C^0$-generic subset of continuous interval maps and a dense subset of metrics topologically equivalent to the Euclidean distance.
We study the rotation sets for homeomorphisms homotopic to the identity on the torus
$\mathbb T^d$
,
$d\ge 2$
. In the conservative setting, we prove that there exists a Baire residual subset of the set
$\text {Homeo}_{0, \lambda }(\mathbb T^2)$
of conservative homeomorphisms homotopic to the identity so that the set of points with wild pointwise rotation set is a Baire residual subset in
$\mathbb T^2$
, and that it carries full topological pressure and full metric mean dimension. Moreover, we prove that for every
$d\ge 2$
the rotation set of
$C^0$
-generic conservative homeomorphisms on
$\mathbb T^d$
is convex. Related results are obtained in the case of dissipative homeomorphisms on tori. The previous results rely on the description of the topological complexity of the set of points with wild historic behavior and on the denseness of periodic measures for continuous maps with the gluing orbit property.
Given a positive integer M and
$q \in (1, M+1]$
we consider expansions in base q for real numbers
$x \in [0, {M}/{q-1}]$
over the alphabet
$\{0, \ldots , M\}$
. In particular, we study some dynamical properties of the natural occurring subshift
$(\boldsymbol{{V}}_q, \sigma )$
related to unique expansions in such base q. We characterize the set of
$q \in \mathcal {V} \subset (1,M+1]$
such that
$(\boldsymbol{{V}}_q, \sigma )$
has the specification property and the set of
$q \in \mathcal {V}$
such that
$(\boldsymbol{{V}}_q, \sigma )$
is a synchronized subshift. Such properties are studied by analysing the combinatorial and dynamical properties of the quasi-greedy expansion of q. We also calculate the size of such classes as subsets of
$\mathcal {V}$
giving similar results to those shown by Blanchard [
10
] and Schmeling in [
36
] in the context of
$\beta $
-transformations.
We describe topological obstructions (involving periodic points, topological entropy and rotation sets) for a homeomorphism on a compact manifold to embed in a continuous flow. We prove that homeomorphisms in a $C^{0}$-open and dense set of homeomorphisms isotopic to the identity in compact manifolds of dimension at least two are not the time-1 map of a continuous flow. Such property is also true for volume-preserving homeomorphisms in compact manifolds of dimension at least five. In the case of conservative homeomorphisms of the torus $\mathbb {T}^{d} (d\ge 2)$ isotopic to identity, we describe necessary conditions for a homeomorphism to be flowable in terms of the rotation sets.
We study the independence density for finite families of finite tuples of sets for continuous actions of discrete groups on compact metrizable spaces. We use it to show that actions with positive naive entropy are Li–Yorke chaotic and untame. In particular, distal actions have zero naive entropy. This answers a question of Lewis Bowen.
Let X be a normal projective variety of dimension n and G an abelian group of automorphisms such that all elements of
$G\setminus \{\operatorname {id}\}$
are of positive entropy. Dinh and Sibony showed that G is actually free abelian of rank
$\le n - 1$
. The maximal rank case has been well understood by De-Qi Zhang. We aim to characterize the pair
$(X, G)$
such that
$\operatorname {rank} G = n - 2$
.
In this paper we analyse the non-wandering set of one-dimensional Greenberg–Hastings cellular automaton models for excitable media with $e\geqslant 1$ excited and $r\geqslant 1$ refractory states and determine its (strictly positive) topological entropy. We show that it results from a Devaney chaotic closed invariant subset of the non-wandering set that consists of colliding and annihilating travelling waves, which is conjugate to a skew-product dynamical system of coupled shift dynamics. Moreover, we determine the remaining part of the non-wandering set explicitly as a Markov system with strictly less topological entropy that also scales differently for large $e,r$.
We study the locally compact abelian groups in the class
${\mathfrak E_{ \lt \infty }}$
, that is, having only continuous endomorphisms of finite topological entropy, and in its subclass
$\mathfrak E_0$
, that is, having all continuous endomorphisms with vanishing topological entropy. We discuss the reduction of the problem to the case of periodic locally compact abelian groups, and then to locally compact abelian p-groups. We show that locally compact abelian p-groups of finite rank belong to
${\mathfrak E_{ \lt \infty }}$
, and that those of them that belong to
$\mathfrak E_0$
are precisely the ones with discrete maximal divisible subgroup. Furthermore, the topological entropy of endomorphisms of locally compact abelian p-groups of finite rank coincides with the logarithm of their scale. The backbone of the paper is the Addition Theorem for continuous endomorphisms of locally compact abelian groups. Various versions of the Addition Theorem are established in the paper and used in the proofs of the main results, but its validity in the general case remains an open problem.
We study the continuity of topological entropy of general diffeomorphisms on line. First, we prove that the entropy map is continuous with respect to the strong $C^{0}$-topology on the union of uniformly topologically hyperbolic diffeomorphisms contained in $\text{Diff}_{0}^{r}(\mathbb{R})$ (whose first derivative is uniformly away from zero), which is a $C^{0}$-open and $C^{r}$-dense subset of $\text{Diff}_{0}^{r}(\mathbb{R})$, $r=1,2,\ldots ,\infty$, and $\unicode[STIX]{x1D714}$ (real analytic). Second, we give some examples where entropy map is not continuous. Finally, we prove some results on the continuity of entropy of general diffeomorphisms on the (real) line.
Using an idea of Doug Lind, we give a lower bound for the Perron–Frobenius degree of a Perron number that is not totally real, in terms of the layout of its Galois conjugates in the complex plane. As an application, we prove that there are cubic Perron numbers whose Perron–Frobenius degrees are arbitrary large, a result known to Lind, McMullen and Thurston. A similar result is proved for bi-Perron numbers.
We investigate the entropy for a class of upper semi-continuous set-valued functions, called Markov set-valued functions, that are a generalization of single-valued Markov interval functions. It is known that the entropy of a Markov interval function can be found by calculating the entropy of an associated shift of finite type. In this paper, we construct a similar shift of finite type for Markov set-valued functions and use this shift space to find upper and lower bounds on the entropy of the set-valued function.
In this paper, it is shown that if a dynamical system is null and distal, then it is equicontinuous. It turns out that a null system with closed proximal relation is mean equicontinuous. As a direct application, it follows that a null dynamical system with dense minimal points is also mean equicontinuous. Meanwhile, a distal system with trivial $\text{Ind}_{\text{fip}}$-pairs and a non-trivial regionally proximal relation of order $\infty$ are constructed.
We study continuous countably (strictly) monotone maps defined on a tame graph, i.e. a special Peano continuum for which the set containing branch points and end points has countable closure. In our investigation we confine ourselves to the countable Markov case. We show a necessary and sufficient condition under which a locally eventually onto, countably Markov map
$f$
of a tame graph
$G$
is conjugate to a map
$g$
of constant slope. In particular, we show that in the case of a Markov map
$f$
that corresponds to a recurrent transition matrix, the condition is satisfied for a constant slope
$e^{h_{\text{top}}(f)}$
, where
$h_{\text{top}}(f)$
is the topological entropy of
$f$
. Moreover, we show that in our class the topological entropy
$h_{\text{top}}(f)$
is achievable through horseshoes of the map
$f$
.
We consider a one-parameter family of dynamical systems $W:[0,1]\rightarrow [0,1]$ constructed from a pair of monotone increasing diffeomorphisms $W_{i}$ such that $W_{i}^{-1}:$$[0,1]\rightarrow [0,1]$$(i=0,1)$. We characterise the set of symbolic itineraries of $W$ using an attractor $\overline{\unicode[STIX]{x1D6FA}}$ of an iterated closed relation, in the terminology of McGehee, and prove that there is a member of the family for which $\overline{\unicode[STIX]{x1D6FA}}$ is symmetrical.
Let
$X$
be a compact, metric and totally disconnected space and let
$f:X\rightarrow X$
be a continuous map. We relate the eigenvalues of
$f_{\ast }:\check{H}_{0}(X;\mathbb{C})\rightarrow \check{H}_{0}(X;\mathbb{C})$
to dynamical properties of
$f$
, roughly showing that if the dynamics is complicated then every complex number of modulus different from 0, 1 is an eigenvalue. This stands in contrast with a classical inequality of Manning that bounds the entropy of
$f$
below by the spectral radius of
$f_{\ast }$
.
By an assignment we mean a mapping from a Choquet simplex
$K$
to probability measure-preserving systems obeying some natural restrictions. We prove that if
$\unicode[STIX]{x1D6F7}$
is an aperiodic assignment on a Choquet simplex
$K$
such that the set of extreme points
$\mathsf{ex}K$
is a countable union
$\bigcup _{n}E_{n}$
, where each set
$E_{n}$
is compact, zero-dimensional and the restriction of
$\unicode[STIX]{x1D6F7}$
to the Bauer simplex
$K_{n}$
spanned by
$E_{n}$
can be ‘embedded’ in some topological dynamical system, then
$\unicode[STIX]{x1D6F7}$
can be ‘realized’ in a zero-dimensional system.
Entropy of categorical dynamics is defined by Dimitrov–Haiden–Katzarkov–Kontsevich. Motivated by the fundamental theorem of the topological entropy due to Gromov–Yomdin, it is natural to ask an equality between the entropy and the spectral radius of induced morphisms on the numerical Grothendieck group. In this paper, we add two results on this equality: the lower bound in a general setting and the equality for orbifold projective lines.