arXiv subjects
Publications and source records attributed to Peter Hazard.
In this paper we show that certain generalizations of the $C^r$-Whitney topology, which include the Hölder-Whitney and Sobolev-Whitney topologies on smooth manifolds, satisfy the Baire property, to wit, the countable intersection of open and dense sets is dense.
We show that generating functions associated to the sequence of convergents of a quadratic irrational are related in a natural way to the dynam- ical zeta function of a hyperbolic automorphism of the 2-torus. As a corollary, this shows that the Lévy constant of a quadratic irrational appears naturally as the topological entropy of such maps.
We show that the generating function corresponding to the sequence of denominators of the best rational approximants of a quadratic irrational is a rational function with integer coefficients. Consequently we can compute the Lévy constant of any quadratic irrational explicitly in terms of a finite number of its convergents.
We give examples of endomorphisms in dimension one with infinite topological entropy which are $α$-Hölder and $(1,p)$-Sobolev for all $0\leqα<1$ and $1\leq p<\infty$. This is constructed within a family of endomorphisms with infinite topological entropy and which traverse all $α$-Hölder and $(1,p)$-Sobolev classes. Finally, we also give examples of endomorphisms, also in dimension one, which lie in the big and little Zygmund classes, answering a question of M. Benedicks.
For bi-Lipschitz homeomorphisms of a compact manifold it is known that topological entropy is always finite. For compact manifolds of dimension two or greater, we show that in the closure of the space of bi-Lipschitz homeomorphisms, with respect to either the Hölder or the Sobolev topologies, topological entropy is generically infinite. We also prove versions of the $C^1$-Closing Lemma in either of these spaces. Finally, we give examples of homeomorphisms with infinite topological entropy which are Hölder and/or Sobolev of every exponent.
In the family of area-contracting Hénon-like maps with zero topological entropy we show that there are maps with infinitely many moduli of stability. Thus one cannot find all the possible topological types for non-chaotic area-contracting Hénon-like maps in a family with finitely many parameters. A similar result, but for the chaotic maps in the family, became part of the folklore a short time after Hénon used such maps to produce what was soon conjectured to be the first non-hyperbolic strange attractor in $\mathbb{R}^2$. Our proof uses recent results about infinitely renormalisable area-contracting Hénon-like maps; it suggests that the number of parameters needed to represent all possible topological types for area-contracting Hénon-like maps whose sets of periods of their periodic orbits are finite (and in particular are equal to $\{1,\, 2,\dots,\,2^{n-1}\}$ or an initial segment of this $n$-tuple) increases with the number of periods. In comparison, among $C^k$-embeddings of the 2-disk with $k\geq 1$, the maximal moduli number for non-chaotic but non area-contracting maps in the interior of the set of zero-entropy is infinite.
We give two general constructions of braid equivalences which exist between certain deformations of the 2-branched Horsehoe map. We then give numerical evidence suggesting that these constructions of braid equivalences are always realised in the Hénon family.
We show that given a one parameter family $F_b$ of strongly dissipative infinitely renormalisable Hénon-like maps, parametrised by a quantity called the `average Jacobian' $b$, the set of all parameters $b$ such that $F_b$ has a Cantor set with unbounded geometry has full Lebesgue measure.