On the Attractor of the Product System

We investigate the attractor of the product system andmainly prove that the likely limit set of the product system equals the product of one of each factor system for n compact systems with solenoid attractors. Specially, this holds for the product map of Feigenbaum maps. Furthermore, we deduce that the Hausdorff dimension of the likely limit set of the product map for Feigenbaummaps is the sum of one of each factor map.


Introduction
An attractor plays a very important role in the studies of dynamical system.The concept of the likely limit set was introduced by Milnor in 1985 (see [1]).As indicated in [1], the likely limit set always exists.Because this kind of attractors gathers the asymptotic behaviors of almost all points, it is very necessary to study them.
In [2], an analytic function similar to a unimodal Feigenbaum map (we call the solution of Feigenbaum functional equation Feigenbaum map) was investigated; the Hausdorff dimension of the likely limit set of it was estimated.We discussed the dynamical properties of unimodal Feigenbaum map, estimated the Hausdorff dimension of the likely limit set for the unimodal Feigenbaum map, and proved that for every  ∈ (0, 1), there always exists a unimodal Feigenbaum map such that the Hausdorff dimension of the likely limit set is .We also considered the kneading sequences of unimodal Feigenbaum maps (see [3]).Similarly to this, we studied the nonunimodal Feigenbaum maps in [4].
In this paper, we explore the dynamics of the product map whose every factor map has solenoid attractor and show that, for this kind of product map, the likely limit set has multiplicative property; that is, the likely limit set of the product map equals the product of one of each factor system.As an application, we consider the product of Feigenbaum maps.The main results are Theorems 8 and 10.

Basic Definitions and Preparations
Milnor introduced the concept of the likely limit set in 1985 [1]. Definition 1.Let  be a compact manifold (with boundary possibly) and  a continuous map of  into itself.The likely limit set Λ = Λ() of  is the smallest closed subset of  with the property that (, ) ⊂ Λ for every point  ∈  outside of a set of Lebesgue measure zero ((, ) denotes the -limit set of the point  under ).
A set  ⊂  is called a minimal set of  if  ̸ =  and (, ) =  for every point  ∈ .As is well known, the minimal set is a nonempty, closed, and invariant subset under , and it has no proper subset with these three properties (see [5]).Therefore, if  is a minimal set with (, ) ⊂  for almost all  ∈ , then  = Λ().
The notion of period interval is an extension of one of period point [6].
⊂  is called a solenoid of  if there are a strictly increasing sequence (  ) ∞ =1 of positive integers and periodic sequences (   ) If a solenoid admits a covering of type (2  ) ∞ =1 , we call it a doubling period solenoid of .
For a proof see [4].
For a proof see [9].

Conclusion
We discussed the likely limit set of product system, show that the likely limit set of product map of two maps with solenoid likely limit sets equals the product of one of each factor map, apply this to the unimodal Feigenbaum map and the nonunimodal Feigenbaum map, and know that the Hausdorff dimension of the likely limit set of the product map of two Feigenbaum maps is the sum of one of each factor map.
and   () is the sum of 2  closed sets which are disjoint from each other; that is,   () is a sequence of closed intervals with period 2  . +1 () is obtained by removing an open interval in each connected component of   () and  ⊃ () ⊃ ⋅ ⋅ ⋅ ⊃   () ⊃ ⋅ ⋅ ⋅ .So Λ() is a doubling period solenoid of .