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Partial variational principle for finitely generated groups of polynomial growth and some foliated spaces

Andrzej Biś (2008)

Colloquium Mathematicae

We generalize the notion of topological pressure to the case of a finitely generated group of continuous maps and introduce group measure entropy. Also, we provide an elementary proof that any finitely generated group of polynomial growth admits a group invariant measure and show that for a group of polynomial growth its measure entropy is less than or equal to its topological entropy. The dynamical properties of groups of polynomial growth are reflected in the dynamics of some foliated spaces.

Periodic billiard orbits in right triangles

Serge Troubetzkoy (2005)

Annales de l’institut Fourier

There is an open set of right triangles such that for each irrational triangle in this set (i) periodic billiards orbits are dense in the phase space, (ii) there is a unique nonsingular perpendicular billiard orbit which is not periodic, and (iii) the perpendicular periodic orbits fill the corresponding invariant surface.

Periodic harmonic functions on lattices and points count in positive characteristic

Mikhail Zaidenberg (2009)

Open Mathematics

This survey deals with pluri-periodic harmonic functions on lattices with values in a field of positive characteristic. We mention, as a motivation, the game “Lights Out” following the work of Sutner [20], Goldwasser- Klostermeyer-Ware [5], Barua-Ramakrishnan-Sarkar [2, 19], Hunzikel-Machiavello-Park [12] e.a.; see also [22, 23] for a more detailed account. Our approach uses harmonic analysis and algebraic geometry over a field of positive characteristic.

Periodic segments and Nielsen numbers

Klaudiusz Wójcik (1999)

Banach Center Publications

We prove that the Poincaré map φ ( 0 , T ) has at least N ( h ˜ , c l ( W 0 W 0 - ) ) fixed points (whose trajectories are contained inside the segment W) where the homeomorphism h ˜ is given by the segment W.

Pourquoi les points périodiques des homéomorphismes du plan tournent-ils autour de certains points fixes ?

Patrice Le Calvez (2008)

Annales scientifiques de l'École Normale Supérieure

Soit f un homéomorphisme du plan qui préserve l’orientation et qui a un point périodique z * de période q ≥ 2 . Nous montrons qu’il existe un point fixe z tel que le nombre d’enlacement de z * et z ne soit pas nul. En d’autres termes, le nombre de rotation de l’orbite de z * dans l’anneau ℝ 2 ∖ { z } est un élément non nul de ℝ / ℤ . Ceci donne une réponse positive à une question posée par John Franks.

Pressure and recurrence

Véronique Maume-Deschamps, Bernard Schmitt, Mariusz Urbański, Anna Zdunik (2003)

Fundamenta Mathematicae

We deal with a subshift of finite type and an equilibrium state μ for a Hölder continuous function. Let αⁿ be the partition into cylinders of length n. We compute (in particular we show the existence of the limit) l i m n → ∞ n - 1 l o g ∑ j = 0 τ ₙ ( x ) μ ( α ⁿ ( T j ( x ) ) ) , where α ⁿ ( T j ( x ) ) is the element of the partition containing T j ( x ) and τₙ(x) is the return time of the trajectory of x to the cylinder αⁿ(x).

Probabilistic cellular automata and random fields with i.i.d. directions

Jean Mairesse, Irène Marcovici (2014)

Annales de l'I.H.P. Probabilités et statistiques

Let us consider the simplest model of one-dimensional probabilistic cellular automata (PCA). The cells are indexed by the integers, the alphabet is { 0 , 1 } , and all the cells evolve synchronously. The new content of a cell is randomly chosen, independently of the others, according to a distribution depending only on the content of the cell itself and of its right neighbor. There are necessary and sufficient conditions on the four parameters of such a PCA to have a Bernoulli product invariant measure....

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