304 research outputs found
Multiple mixing from weak hyperbolicity by the Hopf argument
International audienceWe show that using only weak hyperbolicity (no smoothness, compactness or exponential rates) the Hopf argument produces multiple mixing in an elementary way. While this recovers classical results with far simpler proofs, the point is the broader applicability implied by the weak hypotheses. Some of the results can also be viewed as establishing ''mixing implies multiple mixing'' outside the classical hyperbolic context
The linear request problem
We propose a simple approach to a problem introduced by Galatolo and
Pollicott, consisting in perturbing a dynamical system in order for its
absolutely continuous invariant measure to change in a prescribed way. Instead
of using transfer operators, we observe that restricting to an infinitesimal
conjugacy already yields a solution. This allows us to work in any dimension
and dispense from any dynamical hypothesis. In particular, we don't need to
assume hyperbolicity to obtain a solution, although expansion moreover ensures
the existence of an infinite-dimensional space of solutions.Comment: v2: the approach has been further simplified, only basic differential
calculus is in fact needed instead of basic PD
ORBIT GROWTH OF CONTACT STRUCTURES AFTER SURGERY
Investigation of the effects of a contact surgery construction and of invariance of contact homology reveals a rich new field of inquiry at the intersection of dynamical systems and contact geometry. We produce contact 3-flows not topologically orbit-equivalent to any algebraic flow, including examples on many hyperbolic 3-manifolds, and we show how the surgery produces dynamical complexity for any Reeb flow compatible with the resulting contact structure. This includes exponential complexity when neither the surg-ered flow nor the surgered manifold are hyperbolic. We also demonstrate the use in dynamics of contact homology, a powerful tool in contact geometry
Degeneration of Dynamical Degrees in Families of Maps
The dynamical degree of a dominant rational map
is the quantity
. We study the variation of dynamical
degrees in 1-parameter families of maps . We make a conjecture and ask two
questions concerning, respectively, the set of such that: (1)
; (2) ; (3)
and for "independent"
families of maps. We give a sufficient condition for our conjecture to hold and
prove that it is true for monomial maps. We describe non-trivial families of
maps for which our questions have affirmative and negative answers.Comment: 18 pages. This is an expanded version of the article publishd in Acta
Arithmetica. It contains a corrected statement and full proof of Propostion
11(c
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