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## Réforme du collège: la baisse horaire continue

Volume horaire hebdomadaire actuel/annoncé pour chaque niveau:

• 6ème: 25h + 2h  –> 23h + 3h
• 5ème: 23h + 2h  –> 22h + 4h
• 4ème: 26h + 2h  –> 22h + 4h
• 3ème: 28h30      –> 22h + 4h

Bilan horaire par semaine pour chaque niveau:

• 6ème: – 1h (total) dont  – 2h (disciplinaire) + 1h (accompagnement personnalisé)
• 5ème: +1h (total) dont   -1h (disciplinaire) + 2h (accompagnement personnalisé)
• 4ème: -2h (total) dont   -4h (disciplinaire) + 2h (accompagnement personnalisé)
• 3ème: -2h30 (total) dont -6h30 (disciplinaire) + 4h (interdisciplinaire + accompagnement personnalisé)

Bilan horaire sur les quatre années de collège,

• Total: – 3,2% (3h30 sur 108h30)
• Disciplinaire: -13% (13h30 sur 102h30)

Source: Ministère, Libération

Pour mémoire, horaire obligatoire 1972 (arrêté du 2 mai 1972 d’après ce site)

• 6ème: 27h30
• 5ème: 27h30
• 4ème: 26h + options (latin 4h, grec 3h, langue vivante II 3h, langue vivante I renforcée: 2h)
• 3ème: 26h + options (latin 4h, grec 3h, langue vivante II 3h, langue vivante I renforcée: 2h)
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## C^2 surface diffeomorphisms always have a symbolic extension

Most of topological dynamics studies systems of the form $T:X\to X$ where $T$ is a continuous self-map and $X$ is a compact metric space. One approach is to “reduce” such systems to symbolic dynamical system, i.e., $\sigma:S\to X$ where $S$ is a closed subset of $\{1,\dots,d\}^{\mathbb Z}$ and $\sigma((x_n)_{n\in\mathbb Z})=(x_{n+1})_{n\in\mathbb Z}$ such that $\sigma(S)=S$.

J. Auslander asked about the obstructions for a topological system $T:X\to X$ to have a symbolic extension, i.e., a symbolic system $\sigma:S\to S$ and a continuous surjection $\pi:S\to X$ commuting with the dynamics: $\pi\circ\sigma =T\circ\pi$. There is an obvious one: a symbolic system (and therefore its topological factors) has finite topological entropy. Is there any other?

M. Boyle showed that this was indeed the case. With D. Fiebig and U. Fiebig, he showed that asymptotically h-expansive systems (including $C^\infty$ self-maps of compact manifolds by a result of mine based on Yomdin’s theory) always have a “nice” symbolic extension. T. Downarowicz and S. Newhouse showed that generic $C^1$ map have no symbolic extension whatever, leaving open the question of diffeomorphisms with finite smoothness.

T. Downarowicz and A. Maas showed that $C^r$ interval maps also always have symbolic extensions for $1.

David BURGUET has finally proved the same for arbitrary $C^2$ surface diffeomorphisms, see his preprint here.

Behind these works there is a rich and beautiful topological/ergodic/functional-analytical theory of entropy (called the entropy structure by T. Downarowicz) which does yet have the audience it deserves, in my opinion.

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## Discontinuity of the topological entropy for Lozi maps

I have shown that, like diffeomorphisms, piecewise affine surface homeomorphisms are approximated in entropy by horseshoes, away from their singularities. It follows in particular that their topological entropy is lower-semicontinuous: a small perturbation cannot cause a macroscopic drop in entropy.

The continuity of the entropy for such maps had been an open problem for some time. Rigorous numerical estimates by Duncan SANDS and Yutaka ISHII seemed to suggest some discontinuous drops, but investigation at a small scale suggested these drops to be steep yet continuous variations.

Izzet B. YILDIZ has solved this question by finding for Lozi maps $f_{a,b}(x,y)=(1-a|x|+by,x)$ on $\mathbb R^2$, small numbers $\epsilon_1,\epsilon_2>0$ such that, setting $(a,b)=(1.4+\epsilon_1,0.4+\epsilon_1)$, for all $0<\epsilon<\epsilon_2$:

• $h_{top}(f_{a,b})=0$;
• $h_{top}(f_{a+\epsilon,b})>\frac14\log\frac12(\sqrt{5}+1)$.

The verification turns out to be quite simple (once you know where to look!). The non-wandering set of $f_{a,b}$ is shown to be reduced to be reduced to the fixed points of its fourth iterates, yielding the zero entropy immediately. $f_{a+\epsilon,b}$ on the other hand is shown to admit 2 disjoint closed quadrilaterals $U,V$ such that $f^4(U)$ hyperbolically crosses both $U$ and $V$ and $f^4(V)$ hyperbolically crosses $U$. This means that the sides of $U$ and $V$ can be branded alternatively s and u with the following property. The image of a u side crosses each of $U,V$ it meets, intersecting both their s sides and none of their u sides. This again yields the entropy estimate.

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## Sylvain CROVISIER’s HDR

On November 25th, 2009, Sylvain CROVISIER defended his habilitation à diriger des recherches titled Perturbation de la dynamique de difféomorphismes en petite régularité. He first explained basic perturbation techniques:

• the Anosov-Katok procedure: you use more and more distorted conjugacies such that the limiting dynamics has new properties;
• the closing lemma of Pugh and the subsequent connecting lemmas of Hayashi and Bonatti-Crovisier: you try to glue pieces of orbits while avoiding intermediate visits to the support of the pertubation. This may force you to drop pieces of the orbits. Crovisier nevertheless managed to prove a generalized shadowing lemma constructing orbits guaranteed to visit some neighborhoods.

He then explained how the above was put to work. Together with François Béguin and Frédéric Le Roux, he used the first tool to realize almost arbitrary measurable dynamics as minimal, uniquely ergodic homeomorphisms on arbitray compact manifolds.

With Christian Bonatti, Enrique Pujals and other co-workers, he used the second set of tools (together with constructions of dominated splittings by Wen, Liao-Wen and his own central models) to get deep new results on the dynamics of C^1 generic diffeomorphisms.

Some of the most important ones state that, after a small C^1 perturbation, any diffeomorphism either has a strong global structure (i.e., a phenomenon in the sense of Pujals) or presents a simple obstruction (a mechanism, which one would like to be robust). Namely, up to C^1 perturbations, any diffeomorphism of a compact manifold is:

• Morse-Smale unless it has a  transverse homoclinic intersection (this provides a description of an open and dense subset of the C^1 diffeomorphisms called the weak Palis conjecture);
• partially hyperbolic with a central bundle which is one-dimensional or a sum of two one-dimensional sub-bundles, unless it has a heterodimensional cycle or a homoclinic tangency;
• essentially hyperbolic (hyperbolic from the point of view of its attractors and repellers), unless it has a heterodimensional cycle or a homoclinic tangency.

In answer to the jury’s questions, Sylvain CROVISIER made several additional and more subjective comments:

• It is true that Mañé already characterized the non-hyperbolic diffeomorphisms as those having periodic points that can be made to bifurcate but the goal here is to get robust obstructions;
• It is not clear that dynamics with infinitely many chain recurrent classes can exist robustly: there is no known mechanism for that;
• It seems very difficult to get beyond C^1 with anything like the current techniques – even C^1+1/log seems out of reach;
• It is a reasonable to question to try and develop more precise description of the dynamics especially for the situations that hold on C^1 open sets and therefore occur on C^2 open sets, a usual requisite of ergodic theory techniques;
• The techniques have yielded results very analoguous to Zeeman Tolerance Stability conjecture. However such kinds of stability seem to have more philosophical appeal than a true rôle in the mathematical theory, as opposed to structural stability.
• It is not clear that the above decomposition of the space of C^1 diffeomorphisms extend to higher smoothness (even disregarding the enormous technical difficulties pertaining to the closing lemmas). For instance Bonatti and Diaz hope to show that any diffeomorphism can be C^1 approximated by a hyperbolic one or by one with a homoclinic tangency but this is not the case in the C^2 topology, because of Newhouse phenomenon.
• The techniques seem insufficient to study prevalent or Kolmogorov-typical dynamics.

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## Lower bounds for Hausdorff dimension

A classical theorem (Marstrand 1954) asserts that, given any Borel subset $X\subset\mathbb R^d$, the obvious inequality of the Hausdorff dimensions: $\dim(\pi(X))\leq \min(k,\dim(X))$ is in fact an equality for almost all orthogonal projections $\pi:\mathbb R^d\to\mathbb R^k$. As is often the case it is usually very dificult to prove equality for a given projection.

Preliminary description: Michael HOCHMAN and Pablo SCHMERKIN have posted a preprint presenting a new method for doing this. They actually deal with the stronger statement involving the dimension of measures. The key step is a lower semicontinuity property under an assumption of regularity (the process defined by zooming at typical points must be stationary). The classical a.e. result then yields an open and dense set, which can be controlled using invariance of the considered measures under large groups.

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## Non-convergence des mesures de Gibbs pour T->0

Soit $T:X\to X$ un système dynamique hyperbolique topologiquement transitif muni d’un potentiel continu $\phi:X\to\mathbb R$. La mesure de Gibbs à température $1/\beta$ est par définition l’unique mesure de probabilité $T$-invariante $\mu_\beta$ maximisant l’énergie libre (appelée pression en dynamique…): $h(T,\mu)+\beta\int \phi\, d\mu$$h(T,\mu)$ est l’entropie mesurée.

Il est facile de voir que tout point d’accumulation de $\mu_\beta$ pour $\beta\to\infty$ maximise $\int \phi\, d\mu$. Pour certains systèmes, il existe plusieurs mesures maximisantes. Toutefois, Julien BREMONT a montré en 2001 que les $\mu_\beta$ convergent si $T$ est un sous-décalage de type fini et $\phi$ est localement constant. Depuis, plusieurs chercheurs ont cherché à généraliser ceci au cadre habituel, à savoir $\phi$ hölderienne.

Dans un récent préprint, Jean-René Chazottes et Mike HOCHMAN ont construit un potentiel lipschitzien pour lequel cette convergence n’a pas lieu. Ils montrent de plus que pour les sous-décalages de type fini multi-dimensionnels, cette convergence peut même tomber en défaut pour des potentiels localement constants!

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## Wicked and Weird

Flavio ABDENUR and Martin ANDERSSON: Ergodic theory of generic continuous maps (seminar of F.A. at Parix-XIII).

Both generic endomorphisms of manifolds of any dimension and generic homeomorphisms of manifolds of dimension greater than one exhibit highly pathological ergodic properties with respect to Lebesgue measure: they are weird in the sense that they support neither physical measures nor absolutely continuous measures but at the same time the orbit of Lebesgue almost-every point of M does converge in the Birkhoff sense.

They also study the ergodic properties of homeomorphisms f which are generic within the conjugacy class of expanding maps of the circle. It turns out that the Birkhoff-averaged iterated push-forwards of Lebesgue measure by f accumulate on every Borel invariant probability on M; this implies in particular that the dynamics is wicked: the orbit of Lebesgue-a.e. point of M does NOT converge in the Birkhoff sense.

(adapted from the announcement of the seminar).

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