Shearer’s inequality and infimum rule for Shannon entropy and topological entropy

Downarowicz, Tomasz; Frej, Bartosz; Romagnoli, Pierre Paul

doi:10.1090/conm/669/13423

Cited by 25 publications

(30 citation statements)

References 6 publications

(8 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In the case of Z, Theorem 4.2 in [9] asserts that Definition 2.2 agrees with Definition 2.1. The next fact was proven by Bowen in [7].…”

Section: Additional Notementioning

confidence: 94%

Naive entropy of dynamical systems

Burton

2017

Isr. J. Math.

View full text Add to dashboard Cite

We study an invariant of dynamical systems called naive entropy, which is defined for both measurable and topological actions of any countable group. We focus on nonamenable groups, in which case the invariant is two-valued, with every system having naive entropy either zero or infinity. Bowen has conjectured that when the acting group is sofic, zero naive entropy implies sofic entropy at most zero for both types of systems. We prove the topological version of this conjecture by showing that for every action of a sofic group by homeomorphisms of a compact metric space, zero naive entropy implies sofic entropy at most zero. This result and the simple definition of naive entropy allow us to show that the generic action of a free group on the Cantor set has sofic entropy at most zero. We observe that a distal Γ-system has zero naive entropy in both senses, if Γ has an element of infinite order. We also show that the naive entropy of a topological system is greater than or equal to the naive measure entropy of the same system with respect to any invariant measure.

show abstract

“…In the case of Z, Theorem 4.2 in [9] asserts that Definition 2.2 agrees with Definition 2.1. The next fact was proven by Bowen in [7].…”

Section: Additional Notementioning

confidence: 94%

Naive entropy of dynamical systems

Burton

2017

Isr. J. Math.

View full text Add to dashboard Cite

show abstract

“…A

k

‐cover

(k \in N)

of a set

F \in Fin (G)

is a tuple

(K_{1}, \dots, K_{r})

of elements of

Fin (G)

such that for each

g \in F

the set

{1 ⩽ i ⩽ r : g \in K_{i}}

has at least

k

elements. A function

H : Fin (G) \cup {\emptyset} \to [0, \infty)

: satisfies Shearer's inequality (see ) if for any

F \in Fin (G)

and any

k

‐cover

(K_{1}, \dots, K_{r})

F

, we have

H false(F false) ⩽ false(1 / k false) (H false(K_{1} false) + \dots + H false(K_{r} false))

, is

G

‐invariant if

H (F g) = H (F)

for every

g \in G

and

F \in Fin (G)

, is monotone if for all

A, B \in Fin (G)

wi...…”

Section: Besicovitch Pseudometric and Weyl Pseudometricmentioning

confidence: 99%

“…Let

F \in Fin (G)

and let

(K_{1}, \dots, K_{r})

be a

k -

cover of

F

. Because every element of

F

belongs to

K_{i}

for at least

k

indices

1 ⩽ i ⩽ r

, we have

\begin{matrix} true \\ right \frac{1}{k} (H false(K_{1} false) + \dots + H false(K_{r} false)) & left ⩾ \frac{1}{k} sup_{g \in G} ({normalΔ}_{K_{1} g} false(\underset{̲}{x}, {\underset{̲}{x}}^{'} false) + \dots + {normalΔ}_{K_{r} g} false(\underset{̲}{x}, {\underset{̲}{x}}^{'} false)) \\ left ⩾ \frac{1}{k} sup_{g \in G} (k {normalΔ}_{F g} false(\underset{̲}{x}, {\underset{̲}{x}}^{'} false)) = H (F) . \end{matrix}

By [, Proposition 3.3] every

G

‐invariant, non‐negative function on

Fin (G)

that satisfies Sharer's inequality, obeys the infimum rule.

□

…”

Section: Besicovitch Pseudometric and Weyl Pseudometricmentioning

confidence: 99%

Quasi-uniform convergence in dynamical systems generated by an amenable group action

Łącka

Straszak

2018

J. London Math. Soc.

View full text Add to dashboard Cite

We study the Weyl pseudometric associated with an action of a countable amenable group on a compact metric space. We prove that the topological entropy and the number of minimal subsets of the closure of an orbit are both lower semicontinuous with respect to the Weyl pseudometric. Furthermore, the simplex of invariant measures supported on the orbit closure varies continuously. We apply the Weyl pseudometric to Toeplitz configurations for arbitrary amenable residually finite groups. We introduce the notion of a regular Toeplitz configuration and demonstrate that all regular Toeplitz configurations define minimal and uniquely ergodic systems. We prove that this family is path-connected in the Weyl pseudometric. This leads to a new proof of a theorem of Krieger, saying that Toeplitz configurations can have arbitrary finite entropy.is ergodic if for every G-invariant set A ⊂ X one has either μ(A) = 0 or μ(A) = 1. We denote by † We omit this assumption at the beginning of Section 7. ‡ We used the adjective to distinguish left Følner sequences from their right-and two-sided analogues. In a similar vain notions related to non-commutative groups have 'left' and 'right' variants. For brevity, we will use adjectives 'left/right' only in definitions and routinely omit them later, since the choice of 'left/right' is fixed throughout the paper.

show abstract

“…Motivated by the consideration in [7] for the naive entropy of measure-preserving actions, Burton introduced the naive topological entropy of X in [8]. This is also studied in [11]. For a finite open cover U of X , denote by N (U) the minimal cardinality of subcovers of U.…”

Section: Introductionmentioning

confidence: 99%

“…for U ranging over finite open covers of X . It is known that the naive entropy h nv ( X ) coincides with the classical topological entropy when is amenable [11,Theorem 6.8]. When is sofic, if h nv ( X ) = 0, then the sofic topological entropy of X with respect to any sofic approximation sequence of is either −∞ or 0 (see [8,Theorem 1.1] and [34,Propositions 4.6 and 4.16]).…”

Section: Introductionmentioning

confidence: 99%