Viorel Nitica

Max-plus convex sets and max-plus semispaces. II

We give some complements to Nitica, V. and Singer, I., 2007, Max-plus convex sets and max-plus semispaces, I.Optimization, 56, 171–205. We show that the theories of max-plus convexity in and -convexity in are equivalent, and we deduce some consequences. We show that max-plus convexity in Rn is a multi-order convexity. We give simpler proofs, using only the definition of max-plus segments, of the results of loc. cit. on max-plus semispaces. We show that unlessu2009≤u2009is a total order onA, the results ofloc. cit. on semispaces cannot be generalized in a natural way to the framework ofAn =(An , ≤, ⊗), whereA:=M∪ {−∞}, withM=(M,≤,⊗) being a lattice ordered group and −∞ a “least element’ adjoined toM.

Transitivity of Euclidean extensions of Anosov diffeomorphisms

We consider the class of

Transitivity of Euclidean-type extensions of hyperbolic systems

mathbb{R}^n

Local rigidity of certain partially hyperbolic actions of product type

extensions of Anosov diffeomorphisms on infranilmanifolds, and find necessary and sufficient conditions for topological transitivity. In particular, if the fiber is

A coloring invariant for ribbon L -tetrominos.

mathbb{R}

Stable topological transitivity of skew and principal extensions

, the existence of a semi-orbit with the projection on

An interval version of separation by semispaces in max–min convexity☆

mathbb{R}

Characterization of tropical hemispaces by (P,R)-decompositions

unbounded from above and from below is equivalent to topological transitivity. We also show that in the above class topological transitivity and stable topological transitivity are equivalent.

Revisiting a Tiling Hierarchy

Let f:X→X be the restriction to a hyperbolic basic set of a smooth diffeomorphism. We show that in the class of Cr(r>0) cocycles with fiber the special Euclidean group SE(n), those that are transitive form a residual set (countable intersection of open dense sets). This result is new for odd values of n≥3. More generally, we consider Euclidean-type groups Gn where G is a compact connected Lie group acting linearly on n. When Fix G={0}, it is again the case that the transitive cocycles are residual. When Fix G≠{0}, the same result holds upon restriction to the subset of cocycles that avoid an obvious and explicit obstruction to transitivity.

On the dimension of max-min convex sets

We prove certain rigidity properties of higher–rank abelian product actions of the type α × IdN : Zκ → Diff(M × N), where α is (TNS) (i.e., is hyperbolic and has some special structure of its stable distributions). Together with a result about product actions of property (T ) groups, this implies the local rigidity of higher rank lattice actions of the form α × IdT : Γ → Diff(M × T), provided α has some rigidity properties itself, and contains a (TNS) subaction.