On diffeomorphisms with polynomial growth of the derivative on surfaces

Krzysztof Frączek

Displaying similar documents to “On diffeomorphisms with polynomial growth of the derivative on surfaces”

Exponent of growth of polynomial mappings of $C^{2}$ into $C^{2}$

Jacek Chadzyński, Tadeusz Krasiński (1988)

Banach Center Publications

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Volume of spheres in doubling metric measured spaces and in groups of polynomial growth

Romain Tessera (2007)

Bulletin de la Société Mathématique de France

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Let $G$ be a compactly generated locally compact group and let $U$ be a compact generating set. We prove that if $G$ has polynomial growth, then ${(U^{n})}_{n \in ℕ}$ is a Følner sequence and we give a polynomial estimate of the rate of decay of $\frac{μ (U^{n + 1} ∖ U^{n})}{μ (U^{n})} .$ Our proof uses only two ingredients: the doubling property and a weak geodesic property that we call Property (M). As a matter of fact, the result remains true in a wide class of doubling metric measured spaces including manifolds and graphs. As an application, we obtain...

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 $ϕ \in C ⁰ (ℝ, ℝ^{N})$ .

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...

Explicit bounds for the Łojasiewicz exponent in the gradient inequality for polynomials

Didier D&#039;Acunto, Krzysztof Kurdyka (2005)

Annales Polonici Mathematici

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Let f: ℝⁿ → ℝ be a polynomial function of degree d with f(0) = 0 and ∇f(0) = 0. Łojasiewicz’s gradient inequality states that there exist C > 0 and ϱ ∈ (0,1) such that ${| \nabla f | \geq C | f |}^{ϱ}$ in a neighbourhood of the origin. We prove that the smallest such exponent ϱ is not greater than $1 - R {(n, d)}^{- 1}$ with $R (n, d) = d {(3 d - 3)}^{n - 1}$ .

A set on which the Łojasiewicz exponent at infinity is attained

Jacek Chądzyński, Tadeusz Krasiński (1997)

Annales Polonici Mathematici

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We show that for a polynomial mapping $F = (f ₁, . . ., f ₘ) : ℂ^{n} \to ℂ^{m}$ the Łojasiewicz exponent $_{\infty} (F)$ of F is attained on the set $z \in ℂ^{n} : f ₁ (z) \cdot . . . \cdot f ₘ (z) = 0$ .

Extending piecewise polynomial functions in two variables

Andreas Fischer, Murray Marshall (2013)

Annales de la faculté des sciences de Toulouse Mathématiques

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We study the extensibility of piecewise polynomial functions defined on closed subsets of $ℝ^{2}$ to all of $ℝ^{2}$ . The compact subsets of $ℝ^{2}$ on which every piecewise polynomial function is extensible to $ℝ^{2}$ can be characterized in terms of local quasi-convexity if they are definable in an o-minimal expansion of $ℝ$ . Even the noncompact closed definable subsets can be characterized if semialgebraic function germs at infinity are dense in the Hardy field of definable germs. We also present a piecewise...

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.

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.

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 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.

Heights of squares of Littlewood polynomials and infinite series

Artūras Dubickas (2012)

Annales Polonici Mathematici

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Let P be a unimodular polynomial of degree d-1. Then the height H(P²) of its square is at least √(d/2) and the product L(P²)H(P²), where L denotes the length of a polynomial, is at least d². We show that for any ε > 0 and any d ≥ d(ε) there exists a polynomial P with ±1 coefficients of degree d-1 such that H(P²) < (2+ε)√(dlogd) and L(P²)H(P²)< (16/3+ε)d²log d. A similar result is obtained for the series with ±1 coefficients. Let $A_{m}$ be the mth coefficient of the square f(x)² of...

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...

Entire functions of exponential type not vanishing in the half-plane $ℑ z > k$ , where $k > 0$

Mohamed Amine Hachani (2017)

Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica

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Let $P (z)$ be a polynomial of degree $n$ having no zeros in $| z | < k$ , $k \leq 1$ , and let $Q (z) : = z^{n} \bar{P (1 / \bar{z})}$ . It was shown by Govil that if ${max}_{| z | = 1} | P^{'} (z) |$ and ${max}_{| z | = 1} | Q^{'} (z) |$ are attained at the same point of the unit circle $| z | = 1$ , then $max_{| z | = 1} | P^{'} (z) | \leq \frac{n}{1 + k^{n}} max_{| z | = 1} | P (z) | .$ The main result of the present article is a generalization of Govil’s polynomial inequality to a class of entire functions of exponential type.

On nonsingular polynomial maps of ℝ²

Nguyen Van Chau, Carlos Gutierrez (2006)

Annales Polonici Mathematici

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We consider nonsingular polynomial maps F = (P,Q): ℝ² → ℝ² under the following regularity condition at infinity $(J_{\infty})$ : There does not exist a sequence $(p_{k}, q_{k}) \subset ℂ ²$ of complex singular points of F such that the imaginary parts $(ℑ (p_{k}), ℑ (q_{k}))$ tend to (0,0), the real parts $(ℜ (p_{k}), ℜ (q_{k}))$ tend to ∞ and $F (ℜ (p_{k}), ℜ (q_{k}))) \to a \in ℝ ²$ . It is shown that F is a global diffeomorphism of ℝ² if it satisfies Condition $(J_{\infty})$ and if, in addition, the restriction of F to every real level set $P^{- 1} (c)$ is proper for values of |c| large enough.

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...

Measure-preserving homeomorphisms of noncompact manifolds and mass flow toward ends

Tatsuhiko Yagasaki (2007)

Fundamenta Mathematicae

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Suppose M is a noncompact connected n-manifold and ω is a good Radon measure of M with ω(∂M) = 0. Let ℋ(M,ω) denote the group of ω-preserving homeomorphisms of M equipped with the compact-open topology, and $ℋ_{E} (M, ω)$ the subgroup consisting of all h ∈ ℋ(M,ω) which fix the ends of M. S. R. Alpern and V. S. Prasad introduced the topological vector space (M,ω) of end charges of M and the end charge homomorphism $c^{ω} : ℋ_{E} (M, ω) \to (M, ω)$ , which measures for each $h \in ℋ_{E} (M, ω)$ the mass flow toward ends induced by h. We show that the...

Moving averages

S. V. Butler, J. M. Rosenblatt (2008)

Colloquium Mathematicae

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In ergodic theory, certain sequences of averages $A_{k} f$ may not converge almost everywhere for all f ∈ L¹(X), but a sufficiently rapidly growing subsequence $A_{m_{k}} f$ of these averages will be well behaved for all f. The order of growth of this subsequence that is sufficient is often hyperexponential, but not necessarily so. For example, if the averages are $A_{k} f (x) = 1 / (2^{k}) \sum_{j = 4^{k} + 1}^{4^{k} + 2^{k}} f (T^{j} x)$ , then the subsequence $A_{k ²} f$ will not be pointwise good even on $L^{\infty}$ , but the subsequence $A_{2^{k}} f$ will be pointwise good on L¹. Understanding when the hyperexponential...

Local density of diffeomorphisms with large centralizers

Christian Bonatti, Sylvain Crovisier, Gioia M. Vago, Amie Wilkinson (2008)

Annales scientifiques de l'École Normale Supérieure

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Given any compact manifold $M$ , we construct a non-empty open subset $𝒪$ of the space ${Diff}^{1} (M)$ of $C^{1}$ -diffeomorphisms and a dense subset $𝒟 \subset 𝒪$ such that the centralizer of every diffeomorphism in $𝒟$ is uncountable, hence non-trivial.

$Z ₂^{k}$ -actions with a special fixed point set

Pedro L. Q. Pergher, Rogério de Oliveira (2005)

Fundamenta Mathematicae

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Let Fⁿ be a connected, smooth and closed n-dimensional manifold satisfying the following property: if $N^{m}$ is any smooth and closed m-dimensional manifold with m > n and $T : N^{m} \to N^{m}$ is a smooth involution whose fixed point set is Fⁿ, then m = 2n. We describe the equivariant cobordism classification of smooth actions $(M^{m}; Φ)$ of the group $G = Z ₂^{k}$ on closed smooth m-dimensional manifolds $M^{m}$ for which the fixed point set of the action is a submanifold Fⁿ with the above property. This generalizes a result of F....

An alternative polynomial Daugavet property

Elisa R. Santos (2014)

Studia Mathematica

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We introduce a weaker version of the polynomial Daugavet property: a Banach space X has the alternative polynomial Daugavet property (APDP) if every weakly compact polynomial P: X → X satisfies $m a x_{ω \in} | | I d + ω P | | = 1 + | | P | |$ . We study the stability of the APDP by c₀-, $ℓ_{\infty}$ - and ℓ₁-sums of Banach spaces. As a consequence, we obtain examples of Banach spaces with the APDP, namely $L_{\infty} (μ, X)$ and C(K,X), where X has the APDP.

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 $X$ .

Determination of a type of permutation trinomials over finite fields

Xiang-dong Hou (2014)

Acta Arithmetica

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Let $f = a x + b x^{q} + x^{2 q - 1} \in_{q} [x]$ . We find explicit conditions on a and b that are necessary and sufficient for f to be a permutation polynomial of $_{q ²}$ . This result allows us to solve a related problem: Let $g_{n, q} \in_{p} [x]$ (n ≥ 0, $p = c h a r_{q}$ ) be the polynomial defined by the functional equation $\sum_{c \in_{q}} {(x + c)}^{n} = g_{n, q} (x^{q} - x)$ . We determine all n of the form $n = q^{α} - q^{β} - 1$ , α > β ≥ 0, for which $g_{n, q}$ is a permutation polynomial of $_{q ²}$ .

Hodge type decomposition

Wojciech Kozłowski (2007)

Annales Polonici Mathematici

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In the space $Λ^{p}$ of polynomial p-forms in ℝⁿ we introduce some special inner product. Let $H^{p}$ be the space of polynomial p-forms which are both closed and co-closed. We prove in a purely algebraic way that $Λ^{p}$ splits as the direct sum $d * (Λ^{p + 1}) \oplus δ * (Λ^{p - 1}) \oplus H^{p}$ , where d* (resp. δ*) denotes the adjoint operator to d (resp. δ) with respect to that inner product.