Topological invariants and renormalization of Lorenz maps

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Abstract

We prove that the invariants of the topological semiconjugation of Lorenz maps with β-transformations remains constant on the renormalization archipelagoes and analyze how the dynamics on the archipelagoes depends on its structure.

Introduction

A Lorenz map from an interval I⊂R to itself is a piecewise increasing function f:II with a single discontinuity cI. More precisely:

Definition 1

Let p<c<q. A Lorenz map of class Cr from [p,q] to [p,q] is a pair f=(f,f+), where

  • (1)

    f: [p,c)→[p,q] and f+: (c,q]→[p,q] are strictly increasing maps of class Cr.

  • (2)

    f(p)=p and f+(q)=q.

So f(x)=f(x)ifx<c,f+(x)ifx>c.We denote the set of Lorenz maps of class Cr by Lr.

Typically, Lorenz maps arise as return maps to a cross-section of a semiflow on a two dimensional branched manifold and c is a point from where flow lines never return to I; thus we think of f as being undefined at c.

Lorenz maps have been studied by many mathematicians: for some early work and motivation see [5], [6], [7], [10], [16]. Much of this work considered one topological expanding condition (the pre-images of c are dense in I) which we will not considered here.

In [12], the existence of a semiconjugation from each continuous ℓ-modal map f to a continuous piecewise linear map with |slope|=constant=exp(h(f)) was proved, where h(f) is the topological entropy of f. Moreover, if f is unimodal then h(f) is the only parameter (invariant) that we need to establish the respective piecewise linear map. In the case of Lorenz maps, it is straightforward to extend the result of [12], but it turns out that the corresponding piecewise linear map is also a Lorenz map, i.e. piecewise increasing with a discontinuity at the point λ(f) that depends on f. So unlike in the unimodal case, we will need another parameter (λ(f)) to determine the piecewise linear map semiconjugated to a given Lorenz map f. It turns out that this parameter only depends on the kneading data of f and so it is a topological invariant (just like the topological entropy h(f)).

From the results of [2], it is easy to conclude that if f and g belong to different renormalization domains (see [10] and Section 2.2 below), then (h(f), λ(f))≠(h(g),λ(g)).

In this paper (Theorem 3), we will prove that these two invariants h(f) and λ(f) are constant in the renormalization domains. This complements [2] to conclude that the pair (h(f),λ(f)) classifies topologically (modulus renormalization) the dynamics of Lorenz maps.

In another direction, we describe how the dynamics on a renormalization domain depends on the domain itself (Theorem 2), obtaining in this way a complete description of the combinatorics of the dynamics of Lorenz maps.

Section snippets

Lorenz maps

To avoid technicalities that would complicate the exposition, we will restrict our study to Lorenz maps with negative Schwarzian derivative, i.e. such that both S(f)(x)<0x<c and S(f+)(x)<0x>c, where Sf(x)=f‴(x)f′(x)32f″(x)f′(x)2.We will denote this class of maps by LS. It was proved in [14] that such maps do not have inessential periodic attractors, i.e. each attractive periodic orbit attracts at least one of the critical orbits (the orbits of f(c) and f+(c)). To simplify the exposition,

Semiconjugation

In this section, we will generalize the arguments of [12] to prove that any Lorenz map f with topological entropy h(f)=log(s)>0 is semiconjugated to a piecewise linear map T from the unit interval to itself with constant slope s>1 and with one discontinuity at a point λf(0). Moreover, we will prove that both h(f) and λ(f) are constant in renormalization domains.

The standard definition for the topological entropy of continuous maps using (n,ϵ)-separated sets can be used to define entropy for

Acknowledgements

We thank the referee whose suggestions helped us to reduce considerably the length and also to make more simple the proof of the main theorem.

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