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Hausdorff and packing dimensions for ergodic invariant measures of two-dimensional Lorenz transformations

Franz Hofbauer (2009)

Commentationes Mathematicae Universitatis Carolinae

We extend the notions of Hausdorff and packing dimension introducing weights in their definition. These dimensions are computed for ergodic invariant probability measures of two-dimensional Lorenz transformations, which are transformations of the type occuring as first return maps to a certain cross section for the Lorenz differential equation. We give a formula of the dimensions of such measures in terms of entropy and Lyapunov exponents. This is done for two choices of the weights using the recurrence...

Hausdorff dimension and measures on Julia sets of some meromorphic maps

Krzysztof Barański (1995)

Fundamenta Mathematicae

We study the Julia sets for some periodic meromorphic maps, namely the maps of the form f ( z ) = h ( e x p 2 π i T z ) where h is a rational function or, equivalently, the maps ˜ f ( z ) = e x p ( 2 π i h ( z ) ) . When the closure of the forward orbits of all critical and asymptotic values is disjoint from the Julia set, then it is hyperbolic and it is possible to construct the Gibbs states on J(˜f) for -α log |˜˜f|. For ˜α = HD(J(˜f)) this state is equivalent to the ˜α-Hausdorff measure or to the ˜α-packing measure provided ˜α is greater or smaller than 1....

Heights and totally p-adic numbers

Lukas Pottmeyer (2015)

Acta Arithmetica

We study the behavior of canonical height functions h ̂ f , associated to rational maps f, on totally p-adic fields. In particular, we prove that there is a gap between zero and the next smallest value of h ̂ f on the maximal totally p-adic field if the map f has at least one periodic point not contained in this field. As an application we prove that there is no infinite subset X in the compositum of all number fields of degree at most d such that f(X) = X for some non-linear polynomial f. This answers a...

Hematologic Disorders and Bone Marrow–Peripheral Blood Dynamics

E. Afenya, S. Mundle (2010)

Mathematical Modelling of Natural Phenomena

Hematologic disorders such as the myelodysplastic syndromes (MDS) are discussed. The lingering controversies related to various diseases are highlighted. A simple biomathematical model of bone marrow - peripheral blood dynamics in the normal state is proposed and used to investigate cell behavior in normal hematopoiesis from a mathematical viewpoint. Analysis of the steady state and properties of the model are used to make postulations about the...

Hereditarily non-sensitive dynamical systems and linear representations

E. Glasner, M. Megrelishvili (2006)

Colloquium Mathematicae

For an arbitrary topological group G any compact G-dynamical system (G,X) can be linearly G-represented as a weak*-compact subset of a dual Banach space V*. As was shown in [45] the Banach space V can be chosen to be reflexive iff the metric system (G,X) is weakly almost periodic (WAP). In the present paper we study the wider class of compact G-systems which can be linearly represented as a weak*-compact subset of a dual Banach space with the Radon-Nikodým property. We call such a system a Radon-Nikodým...

Hereditary properties of words

József Balogh, Béla Bollobás (2005)

RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications

Let 𝒫 be a hereditary property of words, i.e., an infinite class of finite words such that every subword (block) of a word belonging to 𝒫 is also in 𝒫 . Extending the classical Morse-Hedlund theorem, we show that either 𝒫 contains at least n + 1 words of length n for every n or, for some N , it contains at most N words of length n for every n . More importantly, we prove the following quantitative extension of this result: if 𝒫 has m ≤ n words of length n then, for every k ≥ n + m , it contains at most ⌈ ( m + 1 ) / 2 ⌉ ⌊ ( m + 1 ) / 2 ⌋ words of length...

Hereditary properties of words

József Balogh, Béla Bollobás (2010)

RAIRO - Theoretical Informatics and Applications

Let P be a hereditary property of words, i.e., an infinite class of finite words such that every subword (block) of a word belonging to P is also in P. Extending the classical Morse-Hedlund theorem, we show that either P contains at least n+1 words of length n for every n or, for some N, it contains at most N words of length n for every n. More importantly, we prove the following quantitative extension of this result: if P has m ≤ n words of length n then, for every k ≥ n + m, it contains at most...

Herman’s last geometric theorem

Bassam Fayad, Raphaël Krikorian (2009)

Annales scientifiques de l'École Normale Supérieure

We present a proof of Herman’s Last Geometric Theorem asserting that if F is a smooth diffeomorphism of the annulus having the intersection property, then any given F -invariant smooth curve on which the rotation number of F is Diophantine is accumulated by a positive measure set of smooth invariant curves on which F is smoothly conjugated to rotation maps. This implies in particular that a Diophantine elliptic fixed point of an area preserving diffeomorphism of the plane is stable. The remarkable...

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