Symbolic extensions in intermediate smoothness on surfaces

David Burguet

Displaying similar documents to “Symbolic extensions in intermediate smoothness on surfaces”

Jumps of entropy for $C^{r}$ interval maps

David Burguet (2015)

Fundamenta Mathematicae

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We study the jumps of topological entropy for $C^{r}$ interval or circle maps. We prove in particular that the topological entropy is continuous at any $f \in C^{r} ([0, 1])$ with $h_{t o p} (f) > (l o g ⁺ | | f^{'} {| |}_{\infty}) / r$ . To this end we study the continuity of the entropy of the Buzzi-Hofbauer diagrams associated to $C^{r}$ interval maps.

Generalized Fokker-Planck equations and convergence to their equilibria

Piotr Biler, Grzegorz Karch (2003)

Banach Center Publications

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We consider extensions of the classical Fokker-Planck equation uₜ + ℒu = ∇·(u∇V(x)) on $ℝ^{d}$ with ℒ = -Δ and V(x) = 1/2|x|², where ℒ is a general operator describing the diffusion and V is a suitable potential.

Normal integral bases and tameness conditions for Kummer extensions

Ilaria Del Corso, Lorenzo Paolo Rossi (2013)

Acta Arithmetica

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We present a detailed analysis of some properties of a general tamely ramified Kummer extension of number fields L/K. Our main achievement is a criterion for the existence of a normal integral basis for a general Kummer extension, which generalizes the existing results. Our approach also allows us to explicitly describe the Steinitz class of L/K and we get an easy criterion for this class to be trivial. In the second part of the paper we restrict to the particular case of tame Kummer...

Relative Bogomolov extensions

Robert Grizzard (2015)

Acta Arithmetica

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A subfield K ⊆ ℚ̅ has the Bogomolov property if there exists a positive ε such that no non-torsion point of $K^{\times}$ has absolute logarithmic height below ε. We define a relative extension L/K to be Bogomolov if this holds for points of $L^{\times} ∖ K^{\times}$ . We construct various examples of extensions which are and are not Bogomolov. We prove a ramification criterion for this property, and use it to show that such extensions can always be constructed if some rational prime has bounded ramification index in K. ...

Greenberg’s conjecture for $ℤ_{p}^{d}$ -extensions

Andrea Bandini (2003)

Acta Arithmetica

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On the joint entropy of $d$ -wise-independent variables

Dmitry Gavinsky, Pavel Pudlák (2016)

Commentationes Mathematicae Universitatis Carolinae

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How low can the joint entropy of $n$ $d$ -wise independent (for $d \geq 2$ ) discrete random variables be, subject to given constraints on the individual distributions (say, no value may be taken by a variable with probability greater than $p$ , for $p < 1$ )? This question has been posed and partially answered in a recent work of Babai [Entropy versus pairwise independence (preliminary version), http://people.cs.uchicago.edu/ laci/papers/13augEntropy.pdf, 2013]. In this paper we improve some...

Topological disjointness from entropy zero systems

Wen Huang, Kyewon Koh Park, Xiangdong Ye (2007)

Bulletin de la Société Mathématique de France

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The properties of topological dynamical systems $(X, T)$ which are disjoint from all minimal systems of zero entropy, $ℳ_{0}$ , are investigated. Unlike the measurable case, it is known that topological $K$ -systems make up a proper subset of the systems which are disjoint from $ℳ_{0}$ . We show that $(X, T)$ has an invariant measure with full support, and if in addition $(X, T)$ is transitive, then $(X, T)$ is weakly mixing. A transitive diagonal system with only one minimal point is constructed. As a consequence, there exists...

ε-Entropy and moduli of smoothness in $L^{p}$ -spaces

A. Kamont (1992)

Studia Mathematica

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The asymptotic behaviour of ε-entropy of classes of Lipschitz functions in $L^{p} (^{d})$ is obtained. Moreover, the asymptotics of ε-entropy of classes of Lipschitz functions in $L^{p} (ℝ^{d})$ whose tail function decreases as $O (λ^{- γ})$ is obtained. In case p = 1 the relation between the ε-entropy of a given class of probability densities on $ℝ^{d}$ and the minimax risk for that class is discussed.

Ergodicity of ℤ² extensions of irrational rotations

Yuqing Zhang (2011)

Studia Mathematica

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Let = [0,1) be the additive group of real numbers modulo 1, α ∈ be an irrational number and t ∈ . We study ergodicity of skew product extensions T : × ℤ² → × ℤ², $T (x, s ₁, s ₂) = (x + α, s ₁ + 2 χ_{[0, 1 / 2)} (x) - 1, s ₂ + 2 χ_{[0, 1 / 2)} (x + t) - 1)$ .

On some nonlinear nonhomogeneous elliptic unilateral problems involving noncontrollable lower order terms with measure right hand side

C. Yazough, E. Azroul, H. Redwane (2013)

Applicationes Mathematicae

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We prove the existence of entropy solutions to unilateral problems associated to equations of the type $A u - d i v (ϕ (u)) = μ \in L ¹ (Ω) + W^{- 1, p^{'} (\cdot)} (Ω)$ , where A is a Leray-Lions operator acting from $W ₀^{1, p (\cdot)} (Ω)$ into its dual $W^{- 1, p (\cdot)} (Ω)$ and $ϕ \in C ⁰ (ℝ, ℝ^{N})$ .

Minimal systems and distributionally scrambled sets

Piotr Oprocha (2012)

Bulletin de la Société Mathématique de France

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In this paper we investigate numerous constructions of minimal systems from the point of view of $(ℱ_{1}, ℱ_{2})$ -chaos (but most of our results concern the particular cases of distributional chaos of type $1$ and $2$ ). We consider standard classes of systems, such as Toeplitz flows, Grillenberger $K$ -systems or Blanchard-Kwiatkowski extensions of the Chacón flow, proving that all of them are DC2. An example of DC1 minimal system with positive topological entropy is also introduced. The above mentioned results...

The topological entropy versus level sets for interval maps (part II)

Jozef Bobok (2005)

Studia Mathematica

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Let f: [a,b] → [a,b] be a continuous function of the compact real interval such that (i) $c a r d f^{- 1} (y) \geq 2$ for every y ∈ [a,b]; (ii) for some m ∈ ∞,2,3,... there is a countable set L ⊂ [a,b] such that $c a r d f^{- 1} (y) \geq m$ for every y ∈ [a,b]∖L. We show that the topological entropy of f is greater than or equal to log m. This generalizes our previous result for m = 2.

Pattern avoidance in partial words over a ternary alphabet

Adam Gągol (2015)

Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica

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Blanched-Sadri and Woodhouse in 2013 have proven the conjecture of Cassaigne, stating that any pattern with $m$ distinct variables and of length at least $2^{m}$ is avoidable over a ternary alphabet and if the length is at least $3 \cdot 2^{m - 1}$ it is avoidable over a binary alphabet. They conjectured that similar theorems are true for partial words – sequences, in which some characters are left “blank”. Using method of entropy compression, we obtain the partial words version of the theorem for ternary words. ...

Gelfand numbers and metric entropy of convex hulls in Hilbert spaces

Bernd Carl, David E. Edmunds (2003)

Studia Mathematica

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For a precompact subset K of a Hilbert space we prove the following inequalities: $n^{1 / 2} c ₙ (c o v (K)) \leq c_{K} (1 + \sum_{k = 1}^{ⁿ} k^{- 1 / 2} e_{k} (K))$ , n ∈ ℕ, and $k^{1 / 2} c_{k + n} (c o v (K)) \leq c [l o g^{1 / 2} (n + 1) ε ₙ (K) + \sum_{j = n + 1}^{\infty} ε_{j} (K) / (j l o g^{1 / 2} (j + 1))]$ , k,n ∈ ℕ, where cₙ(cov(K)) is the nth Gelfand number of the absolutely convex hull of K and $ε_{k} (K)$ and $e_{k} (K)$ denote the kth entropy and kth dyadic entropy number of K, respectively. The inequalities are, essentially, a reformulation of the corresponding inequalities given in [CKP] which yield asymptotically optimal estimates of the Gelfand numbers cₙ(cov(K)) provided that the entropy numbers εₙ(K)...

The power set of ω Elementary submodels and weakenings of CH

István Juhász, Kenneth Kunen (2001)

Fundamenta Mathematicae

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We define a new principle, SEP, which is true in all Cohen extensions of models of CH, and explore the relationship between SEP and other such principles. SEP is implied by each of CH*, the weak Freeze-Nation property of (ω), and the (ℵ₁,ℵ₀)-ideal property. SEP implies the principle $C ₂^{s} (ω ₂)$ , but does not follow from $C ₂^{s} (ω ₂)$ , or even $C^{s} (ω ₂)$ .

Class group of a cyclotomic $ℤ_{p} \times ℤ_{ℓ}$ -extension

Humio Ichimura (2011)

Acta Arithmetica

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Measures of maximal entropy for random $β$ -expansions

Karma Dajani, Martijn de Vries (2005)

Journal of the European Mathematical Society

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Let $β > 1$ be a non-integer. We consider $β$ -expansions of the form $\sum_{i = 1}^{\infty} d_{i} / β^{i}$ , where the digits ${(d_{i})}_{i \geq 1}$ are generated by means of a Borel map $K_{β}$ defined on ${0, 1}^{ℕ} \times [0, ⌊ β ⌋ / (β - 1)]$ . We show that $K_{β}$ has a unique mixing measure $ν_{β}$ of maximal entropy with marginal measure an infinite convolution of Bernoulli measures. Furthermore, under the measure $ν_{β}$ the digits ${(d_{i})}_{i \geq 1}$ form a uniform Bernoulli process. In case 1 has a finite greedy expansion with positive coefficients, the measure of maximal entropy is Markov. We also discuss the uniqueness...

On the directional entropy of ℤ²-actions generated by cellular automata

M. Courbage, B. Kamiński (2002)

Studia Mathematica

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We show that for any cellular automaton (CA) ℤ²-action Φ on the space of all doubly infinite sequences with values in a finite set A, determined by an automaton rule $F = F_{[l, r]}$ , l,r ∈ ℤ, l ≤ r, and any Φ-invariant Borel probability measure, the directional entropy $h_{v ⃗} (Φ)$ , v⃗= (x,y) ∈ ℝ², is bounded above by $m a x (| z_{l} |, | z_{r} |) l o g A$ if $z_{l} z_{r} \geq 0$ and by $| z_{r} - z_{l} |$ in the opposite case, where $z_{l} = x + l y$ , $z_{r} = x + r y$ . We also show that in the class of permutative CA-actions the bounds are attained if the measure considered is uniform Bernoulli.

Preservation of properties of a map by forcing

Akira Iwasa (2022)

Commentationes Mathematicae Universitatis Carolinae

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Let $f : X \to Y$ be a continuous map such as an open map, a closed map or a quotient map. We study under what circumstances $f$ remains an open, closed or quotient map in forcing extensions.

Interaction between cellularity of Alexandroff spaces and entropy of generalized shift maps

Fatemah Ayatollah Zadeh Shirazi, Sahar Karimzadeh Dolatabad, Sara Shamloo (2016)

Commentationes Mathematicae Universitatis Carolinae

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In the following text for a discrete finite nonempty set $K$ and a self-map $ϕ : X \to X$ we investigate interaction between different entropies of generalized shift $σ_{ϕ} : K^{X} \to K^{X}$ , ${(x_{α})}_{α \in X} \mapsto {(x_{ϕ (α)})}_{α \in X}$ and cellularities of some Alexandroff topologies on $X$ .

Entropy and approximation numbers of embeddings between weighted Besov spaces

Iwona Piotrowska (2008)

Banach Center Publications

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The present paper is devoted to the study of the “quality” of the compactness of the trace operator. More precisely, we characterize the asymptotic behaviour of entropy numbers of the compact map $t r_{Γ} : B_{p ₁, q}^{s} (ℝ ⁿ, w_{ϰ}^{Γ}) \to L_{p ₂} (Γ)$ , where Γ is a d-set with 0 < d < n and $w_{ϰ}^{Γ}$ a weight of type $w_{ϰ}^{Γ} (x) d i s t {(x, Γ)}^{ϰ}$ near Γ with ϰ > -(n-d). There are parallel results for approximation numbers.

Some logarithmic function spaces, entropy numbers, applications to spectral theory

Haroske Dorothee

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AbstractIn [18] and [19] we have studied compact embeddings of weighted function spaces on ℝⁿ, $i d : H_{q}^{s} (w (x), ℝ ⁿ) \to L ₚ (ℝ ⁿ)$ , s>0, 1 < q ≤ p< ∞, s-n/q+n/p > 0, with, for example, $w (x) = ⟨ x ⟩^{α}$ , α > 0, or $w (x) = l o g^{β} ⟨ x ⟩$ , β > 0, and $⟨ x ⟩ = {(2 + | x | ²)}^{1 / 2}$ . We have determined the behaviour of their entropy numbers eₖ(id). Now we are interested in the limiting case 1/q = 1/p + s/n. Let $w (x) = l o g^{β} ⟨ x ⟩$ , β > 0. Our results in [18] imply that id cannot be compact for any β > 0, but after replacing the target space Lₚ(ℝⁿ) by a “slightly” larger one, $L ₚ {(l o g L)}_{- a} (ℝ ⁿ)$ , a...

Hyperbolic measure of maximal entropy for generic rational maps of $ℙ^{k}$

Gabriel Vigny (2014)

Annales de l’institut Fourier

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Let $f$ be a dominant rational map of $ℙ^{k}$ such that there exists $s < k$ with $λ_{s} (f) > λ_{l} (f)$ for all $l$ . Under mild hypotheses, we show that, for $A$ outside a pluripolar set of $Aut (ℙ^{k})$ , the map $f \circ A$ admits a hyperbolic measure of maximal entropy $log λ_{s} (f)$ with explicit bounds on the Lyapunov exponents. In particular, the result is true for polynomial maps hence for the homogeneous extension of $f$ to $ℙ^{k + 1}$ . This provides many examples where non uniform hyperbolic dynamics is established. One of the key tools is to approximate...

On the compositum of all degree $d$ extensions of a number field

Itamar Gal, Robert Grizzard (2014)

Journal de Théorie des Nombres de Bordeaux

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We study the compositum $k^{[d]}$ of all degree $d$ extensions of a number field $k$ in a fixed algebraic closure. We show $k^{[d]}$ contains all subextensions of degree less than $d$ if and only if $d \leq 4$ . We prove that for $d > 2$ there is no bound $c = c (d)$ on the degree of elements required to generate finite subextensions of $k^{[d]} / k$ . Restricting to Galois subextensions, we prove such a bound does not exist under certain conditions on divisors of $d$ , but that one can take $c = d$ when $d$ is prime. This question was inspired by work of...