An invitation to rough dynamics zipper maps Benoît R. KloecknerNicolae Mihalache

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An invitation to rough dynamics:

zipper maps

Benoît R. Kloeckner *Nicolae Mihalache ∗

October 25, 2022

In the ﬁeld of dynamical systems, it is not rare to meet irregular functions,

which are typically Hölder but not Lipschitz (e.g. the Weierstrass functions).

Our goal is to scratch the surface of the following question: what happens if

we consider irregular maps and iterate them?

We introduce the family of zipper maps, which are irregular in the above

sense, and study some of their dynamical properties. For a large set of

parameters, the corresponding zipper map admits horseshoe of all orders; as

an immediate consequence, every order on 𝑘ℓ points can be realized by 𝑘

orbits of length ℓof the map.

These maps have inﬁnite topological entropy, and we reﬁne this statement

by showing that they have positive metric mean dimension with respect

to the Euclidean metric, as well as by introducing other notions of higher

complexity.

Finally, we prove that every interval map (thus including zipper maps)

have vanishing absolute metric mean dimension, proving a small case of

the conjecture that the absolute metric mean dimension coincides with the

topological mean dimension.

Contents

1 Introduction 2

2 Basic properties of zipper maps 8

3 Horseshoes of arbitrary order 10

4 Relative metric mean dimension and other high-complexity measurements 17

∗Univ Paris Est Creteil, Univ Gustave Eiﬀel, CNRS, LAMA UMR8050, F-94010 Creteil, France

arXiv:2210.13038v1 [math.DS] 24 Oct 2022

5 Vanishing of the absolute metric mean dimension for interval maps 22

1 Introduction

Dynamical systems form by now an incredibly broad and deep ﬁeld, with many diﬀerent

kinds of systems having been studied thoroughly. Even restricting to continuous maps

acting on a compact space, one could spend a lifetime learning about them. However, it

seems that there is still a blind spot in the literature: maps of low regularity, by which

we mean less than locally lipschitz, typically not more than Hölder-continuous. The goal

of this article is to showcase the kind of dynamical properties that one can observe in

one speciﬁc family of such irregular maps.

1.1 Zipper maps

Let us ﬁrst introduce the family of maps we shall be interested in. Their graph will be a

zipper curve (hence the name); as functions, they where introduced by Bruneau [Bru74]

as extremal points in certain functional spaces; until now they seem not to have been

considered as maps that one can iterate. They have the advantage of being relatively

explicit while exhibiting quite wild dynamical properties.

By C𝛼([0,1]) we mean the Banach space of 𝛼-Hölder functions deﬁned on [0,1] with

values in Rendowed with its usual norm

‖𝑓‖𝛼= sup

𝑥̸=𝑦

|𝑓(𝑥)−𝑓(𝑦)|

|𝑥−𝑦|𝛼+ sup

𝑥|𝑓(𝑥)|

and by C𝛼

0the convex, closed subset of the functions 𝑓with range [0,1] and such that

𝑓(0) = 0 and 𝑓(1) = 1 (which shall then be seen as point-valued maps rather than

scalar-valued functions). When 𝛼= 0 we mean the space of continuous functions or

maps, endowed with the supremum norm.

Let 𝑝=(𝑥1, 𝑦1),(𝑥2, 𝑦2)∈(0,1)2be a pair of points in the unit square such that

𝑥2> 𝑥1and 𝑦2< 𝑦1, and let Φ𝑝:C0

0→C0

0be the map deﬁned by

Φ𝑝𝑓(𝑥) =

⎧

⎪

⎨

⎪

⎩

𝑦1𝑓𝑥

𝑥1if 𝑥∈[0, 𝑥1]

𝑦1−(𝑦1−𝑦2)𝑓𝑥−𝑥1

𝑥2−𝑥1if 𝑥∈[𝑥1, 𝑥2]

𝑦2+ (1 −𝑦2)𝑓𝑥−𝑥2

1−𝑥2if 𝑥∈[𝑥2,1]

Then Φ𝑝is a contraction in the uniform norm, of ratio

𝑣max = max(𝑦1, 𝑦1−𝑦2,1−𝑦2)<1,

and thus has a unique ﬁxed point 𝑍𝑝∈C0

0, which we shall call the zipper map of

parameter 𝑝; the case 𝑝=(.3, .7),(.8, .1)is shown in Figure 1.

Figure 1: The graph of a zipper map. Note that 1is mapped to 1, which is barely visible

given the extremely high speed of variation at that point.

We will further restrict to the hypersensitive case where the (piecewise aﬃne) image

of the identity map by Φ𝑝has all its slopes greater than 1in absolute value:

1< 𝜆min := min 𝑦1

𝑥1

,𝑦1−𝑦2

𝑥2−𝑥1

,1−𝑦2

1−𝑥2(1)

1.2 Main results

Topological entropy is one of the paradigmatic measurement of chaos, and shifts on ﬁnite

alphabets are among the most basic models for chaotic maps; as we will recall below,

for continuous interval maps they are strongly related through “horseshoes”. Our ﬁrst

result shows that many zipper maps exhibit a strong form of chaos by having horseshoes

of arbitrarily order.

Theorem A. Let 𝑇=𝑍𝑝be an hypersensitive zipper map and additionally assume

either one of these conditions:

i. 𝑝is symmetric with respect to the center of the square, i.e. 𝑥1+𝑥2=𝑦1+𝑦2= 1,

ii. the pair of second coordinates (𝑦1, 𝑦2)of 𝑝lies in the open set

𝐵={(𝑦1, 𝑦2)∈(0,1)2|𝑦2

1> 𝑦2and 𝑦1>(2 −𝑦2)𝑦2}.

Then 𝑇admits horseshoes of all orders.

Figure 2: The domain 𝐵for assumption ii in Theorem A.

The conclusion means that for all 𝑘∈Nthere exist compact sub-intervals 𝐼1, . . . , 𝐼𝑘

of [0,1] with pairwise disjoint interiors, such that 𝑇(𝐼𝑖)⊃𝐼𝑗for all 𝑖, 𝑗 ∈ {1, . . . , 𝑘}. We

can also get pairwise disjoint intervals, simply by taking 2𝑘instead of 𝑘and keep every

other interval in the family, in the linear order of [0,1], see Section 3.3.

It is known [Mis79] (see also [Rue17] and references therein) that having inﬁnite topo-

logical entropy is equivalent to having horseshoes of arbitrary size 𝑘in iterates 𝑇𝑜(log 𝑘);

Theorem Ashow that some zipper map possess a much stronger property, in that we do

not need to iterate them to obtain horseshoes of all order.

The presence of these horseshoes easily imply some “universality” properties.

Corollary B. For all hypersensitive zipper map 𝑇=𝑍𝑝satisfying either assumptions

of Theorem Aand all 𝑘, ℓ ∈N, every total order on 𝑘×ℓsymbols is realised by 𝑘orbits

of length ℓof 𝑇.

This means that for all total strict order ≺on the symbols (𝑠𝑗

𝑖)0≤𝑗≤ℓ

1≤𝑖≤𝑘

, there exist

𝑥1, . . . , 𝑥𝑘∈[0,1] such that for all 𝑖, 𝑗, 𝑖′, 𝑗′:

𝑠𝑗

𝑖≺𝑠𝑗′

𝑖′⇔𝑇𝑗𝑥𝑖< 𝑇 𝑗′𝑥𝑖′.

Let Ω⊆R,𝑆: Ω →Ωand 𝑇: [0,1] →[0,1]. We say that 𝑆is embedded in 𝑇(as a

dynamical system) if there exists 𝜋: Ω →[0,1], injective, such that

𝜋∘𝑆=𝑇∘𝜋.

Corollary C. Let 𝑇=𝑍𝑝be a hypersensitive zipper map satisfying either assumption

of Theorem Aand Ωa ﬁnite set. Then any map 𝑆: Ω →Ωis embedded in 𝑇.

The existence of horseshoe of arbitrary order implies that zipper maps to which The-

orem Aapplies have inﬁnite topological entropy, a fact easy to establish directly for

all hypersensitive zipper maps. But we can say more, by using a variation of entropy

suitable for highly chaotic systems: metric mean dimension, whose deﬁnition is recalled

in Section 4.

Theorem D. Every hypersensitive zipper map 𝑇=𝑍𝑝has positive metric mean dimen-

sion relative to the Euclidean metric:

mdim𝑀(𝑇, |·|)≥log 𝜆min

|log ℎmin|>0,

where ℎmin = min(𝑥1, 𝑥2−𝑥1,1−𝑥2)<1and 𝜆min is deﬁned in (1).

Corollary E. Every hypersensitive zipper map 𝑇=𝑍𝑝admits an invariant probability

measure 𝜇with positive Kolmogorov-Sinai mean dimension.

The Kolmogorov-Sinai mean dimension is an invariant of a measured dynamical sys-

tem (𝑇, 𝜇)which we introduce in Section 4. Its positivity means that for arbitrarily high

𝑘, there exist measurable partitions of [0,1] into 𝑘subsets for which the entropy grows

as a multiple of log 𝑘as 𝑘→ ∞. Corollary Eis deduced from Theorem Dthrough an

inequality providing one half of a variational principle, Theorem 4.2: the existence of a

metric of ﬁnite dimension for which the relative metric mean dimension is positive en-

ables the construction of measures of positive Kolmogorov-Sinai metric mean dimension.

However, unlike the classical variational principle, the complexity of these measures does

not bound below the metric mean dimension of arbitrary metrics, even when controlling

their dimension:

Theorem F. Every continuous map 𝑇: [0,1] →[0,1] has zero absolute metric mean

dimension; more precisely, there exist a metric 𝑑on [0,1], inducing the usual topology,

such that mdim𝑀(𝑇, 𝑑) = 0 and dim+

𝑀([0,1], 𝑑)=1.

Here dim+

𝑀denotes the upper Minkowski dimension; it bounds from above the Haus-

dorﬀ dimension, which must thus also be equal to 1. This result shows a small case of

the conjecture (see e.g. [LT19]) that mean dimension equals the inﬁmum over all metrics

of the metric mean dimension.

1.3 Further directions

1.3.1 Further properties of zipper maps

We have left many questions open even when restricting to the case of zipper maps. One

could ﬁrst determine the exact range of Theorem A, by determining the set of all values

of the parameter 𝑝for which 𝑇=𝑍𝑝admits horseshoes of all order.

Another intriguing question is that of topological conjugacy: for which 𝑝, 𝑝′are the

zipper maps 𝑍𝑝,𝑍𝑝′topologically conjugate? Is there a continuum of conjugacy classes

among the 𝑍𝑝? Note that one does not hope for any sort of topological stability here: the

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Aninvitationtoroughdynamics:zippermapsBenoîtR.Kloeckner*NicolaeMihalache*October25,2022Inthefieldofdynamicalsystems,itisnotraretomeetirregularfunctions,whicharetypicallyHölderbutnotLipschitz(e.g.theWeierstrassfunctions).Ourgoalistoscratchthesurfaceofthefollowingquestion:whathappensifweconsiderirregu...

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An invitation to rough dynamics zipper maps Benoît R. KloecknerNicolae Mihalache

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