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On gaps in Rényi β -expansions of unity for β > 1 an algebraic number

Jean-Louis Verger-Gaugry (2006)

Annales de l’institut Fourier

Let β > 1 be an algebraic number. We study the strings of zeros (“gaps”) in the Rényi β -expansion   d β ( 1 ) of unity which controls the set ℤ β of β -integers. Using a version of Liouville’s inequality which extends Mahler’s and Güting’s approximation theorems, the strings of zeros in d β ( 1 ) are shown to exhibit a “gappiness” asymptotically bounded above by   log ( M ( β ) ) / log ( β ) , where   M ( β )   is the Mahler measure of   β . The proof of this result provides in a natural way a new classification of algebraic numbers > 1 with classes called Q...

On gradient-like random dynamical systems

Aya Hmissi, Farida Hmissi, Mohamed Hmissi (2012)

ESAIM: Proceedings

This paper deals with some characterizations of gradient-like continuous random dynamical systems (RDS). More precisely, we establish an equivalence with the existence of random continuous section or with the existence of continuous and strict Liapunov function. However and contrary to the deterministic case, parallelizable RDS appear as a particular case of gradient-like RDS.The obtained results are generalizations of well-known analogous theorems in the framework of deterministic dynamical systems....

On heredity of strongly proximal actions

C. Robinson Edward Raja (2003)

Archivum Mathematicum

We prove that action of a semigroup T on compact metric space X by continuous selfmaps is strongly proximal if and only if T action on 𝒫 ( X ) is strongly proximal. As a consequence we prove that affine actions on certain compact convex subsets of finite-dimensional vector spaces are strongly proximal if and only if the action is proximal.

On implicit Lagrangian differential systems

S. Janeczko (2000)

Annales Polonici Mathematici

Let (P,ω) be a symplectic manifold. We find an integrability condition for an implicit differential system D' which is formed by a Lagrangian submanifold in the canonical symplectic tangent bundle (TP,ὡ).

On inertial manifolds for reaction-diffusion equations on genuinely high-dimensional thin domains

M. Prizzi, K. P. Rybakowski (2003)

Studia Mathematica

We study a family of semilinear reaction-diffusion equations on spatial domains Ω ε , ε > 0, in ℝ l lying close to a k-dimensional submanifold ℳ of ℝ l . As ε → 0⁺, the domains collapse onto (a subset of) ℳ. As proved in [15], the above family has a limit equation, which is an abstract semilinear parabolic equation defined on a certain limit phase space denoted by H ¹ s ( Ω ) . The definition of H ¹ s ( Ω ) , given in the above paper, is very abstract. One of the objectives of this paper is to give more manageable characterizations...

On inhomogeneous self-similar measures and their L q spectra

Przemysław Liszka (2013)

Annales Polonici Mathematici

Let S i : ℝ d → ℝ d for i = 1,..., N be contracting similarities, let ( p ₁ , . . . , p N , p ) be a probability vector and let ν be a probability measure on ℝ d with compact support. It is well known that there exists a unique inhomogeneous self-similar probability measure μ on ℝ d such that μ = ∑ i = 1 N p i μ ∘ S i - 1 + p ν . We give satisfactory estimates for the lower and upper bounds of the L q spectra of inhomogeneous self-similar measures. The case in which there are a countable number of contracting similarities and probabilities is considered. In particular, we...

On invariant measures for the tend map.

Francesc Bofill (1988)

Stochastica

The bifurcation structure of a one parameter dependent piecewise linear population model is described. An explicit formula is given for the density of the unique invariant absolutely continuous probability measure mub for each parameter value b. The continuity of the map b --> mub is established.

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