Topological entropy and diffeomorphisms of surfaces with wandering domains

Abstract

Abstract. Let M be a closed surface and f a diffeomorphism of M . A diffeomorphism is said to permute a dense collection of domains, if the union of the domains are dense and the iterates of any one domain are mutually disjoint. In this note, we show that if f ∈ Diff 1+α (M ), with α > 0, and permutes a dense collection of domains with bounded geometry, then f has zero topological entropy. Definitions and statement of resultsA result of Norton and Sullivan [8] states that a diffeomorphism f ∈ Diff 3 0 (T 2 ) having Denjoy-type can not have a wandering disk whose iterates have the same generic shape. By diffeomorphisms of Denjoy-type are meant diffeomorphisms of the two-torus, isotopic to the identity, that are obtained as an extension of an irrational translation of the torus, for which the semi-conjugacy has countably many non-trivial fibers. If these fibers have non-empty interior, then the corresponding diffeomorphism has a wandering disk. Further, by generic shape is meant that the only elements of SL(2, Z) preserving the shape are elements of SO(2, Z), such as round disks and squares. In a similar spirit, Bonatti, Gambaudo, Lion and Tresser in [1] show that certain infinitely renormalizable diffeomorphisms of the two-disk that are sufficiently smooth, can not have wandering domains if these domains have a certain boundedness of geometry.In this note, we study an analogous problem, namely the interplay between the geometry of iterates of domains under a diffeomorphism and its topological entropy.To state the precise result, we first need some definitions. Let (M, g) be a closed surface, that is, a smooth, closed, oriented Riemannian two-manifold, equipped with the canonical metric g induced from the standard conformal metric of the universal cover P Given f ∈ Homeo(M ), for each n ≥ 1, define the metric Ferry Kwakkel and Vladimir Markovic(n, ) separated if d n (x, y) ≥ for every x, y ∈ U with x = y. Let N (n, ) be the maximum cardinality of an (n, ) separated set. The topological entropy is defined asNext, we make precise the notion of a homeomorphism of a surface permuting a dense collection of domains. Note that we do not assume a domain to be recurrent, nor do we assume the orbit of a single domain to be dense. A wandering domain is a domain with mutually disjoint iterates under f such that the orbit of the domain is recurrent. Thus a diffeomorphism with a wandering domain with dense orbit is a special case of definition 1.1. Denote exp p : T p M → M the exponential mapping at p ∈ M . The injectivity radius at a point p ∈ M is defined as the largest radius for which exp p is a diffeomorphism. The injectivity radius ι(M ) of M is the infimum of the injectivity radii over all points where B(p, r) ⊂ M is the ball centered at p ∈ M with radius r > 0. If no such β exists, then the collection is said to have unbounded geometry.By Cl(D k ) being contractible in M we mean that Cl(D k ) is contained in an embedded topological disk in M . Our definition of bounded geometry is equivalent to the notion of bounded g...

Takens' last problem and existence of non-trivial wandering domains

“…However, it can not be extended to any C 2 as well as C 1 diffeomorphism whose derivative is a function of bounded variation, see in [8]. Subsequently, similar phenomena for high dimensional diffeomorphisms were studied by several authors, for example [22,18,30,4,31,27]. Also, for unimodal as well as multimodal maps on an interval or a circle, the main difficulty in their classification in real analytic category was to show the absence of wandering domains, which were developed by many dynamicists [7,28,2,8,40], see the survey of van Strien [39].…”

No bounded geometry wandering domains for sufficiently regular automorphisms

“…More generally, let (M, g) be a closed, i.e., compact and with no boundary, Riemannian n-manifold. A domain D in a Riemannian manifold M is an open connected set whose closure D is assumed to be an embedded closed topological ball that is contractible in M. Following terminology of F. Kwakkel and V. Markovic [KM10], we say a homeomorphism f of M permutes a dense collection of domains if there exists a connected, completely invariant, and nowhere dense compact subset Λ ⊂ M such that the collection {D i } i∈I of connected components of the complement M \ Λ has the following properties: for each i ∈ I, the domain D i is the interior of its closure D i , the closures D i , i ∈ I, are pairwise disjoint, and for each…”

“…In the spirit of lowering regularity, in [KM10], F. Kwakkel and V. Markovic posed the following question. Question 1.9 (F. Kwakkel, V. Markovic [KM10]).…”

“…In the spirit of lowering regularity, in [KM10], F. Kwakkel and V. Markovic posed the following question. Question 1.9 (F. Kwakkel, V. Markovic [KM10]). If M is a closed Riemann surface equipped with the canonical metric induced from the standard conformal metric of the universal cover, do there exist diffeomorphisms f ∈ Diff 1 (M) with positive topological entropy that permute a dense collection of domains with bounded geometry?…”

Nonexistence of wandering domains for strongly dissipative infinitely renormalizable Hénon maps at the boundary of chaos

“…A relevant work by Kiriki and Soma [42] found Hénon-like maps having wandering domains by using a homoclinic tangency of some saddle fixed point [41,43]. On the other hand, there are studies [45,46,57,60] suggesting that some types of systems may have no wandering domains. However, this has been proved in just some such systems [6,58].…”