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On Hausdorff dimension of invariant sets for expanding maps of a circle
| Content Provider | Semantic Scholar |
|---|---|
| Author | Urba'nski, Mariusz |
| Copyright Year | 1986 |
| Abstract | Given an orientation preserving C expanding mapping g: S^* S of a circle we consider the family of closed invariant sets Kg(e) denned as those points whose forward trajectory avoids the interval (0, e). We prove that topological entropy of g\Kg(e) is a Cantor function of e. If we consider the map g(z) = z q then the Hausdorff dimension of the corresponding Cantor set around a parameter e in the space of parameters is equal to the Hausdorff dimension of Kg(e). In § 3 we establish some relationships between the mappings g\Kg(e) and the theory of j3-transformations, and in the last section we consider DE-bifurcations related to the sets Kg(e). 0. Introduction First we give the following: Definition 1. Let g: S -* S be a C expanding map (i.e. such that there exists n > 1 for which |(/")'(*)l> 1 for every x e S) which preserves orientation. Let 0< e < 1 and let (0, e) denote the open interval on S of length e whose left endpoint is one of the fixed points for g. We choose an orientation and suppose that the whole length of S is equal to 1. Now we define the set Kg(E)=r)g(S\(Q,e)). It is easy to see that Kg(e) is a closed, invariant set for g, that is, g(Kg(e))c Kg(e) and furthermore g(Kg(e)) = Kg(e). However, we remark that the inclusion g~\Kg(e))<= Kg{e) does not hold except for Kg(e) = 0 or S . Let {KA}AeA be a continuous family of mixing repellers for a real analytic family {/A : S 1 -» S'JxeA of real analytic mappings and let { R}A£A be a real analytic family of real analytic functions. Then as Ruelle [R] proved, the pressure function A3A->P/A|KA( |
| Starting Page | 295 |
| Ending Page | 309 |
| Page Count | 15 |
| File Format | PDF HTM / HTML |
| Volume Number | 6 |
| Alternate Webpage(s) | https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0143385700003461 |
| Alternate Webpage(s) | https://doi.org/10.1017/S0143385700003461 |
| Language | English |
| Access Restriction | Open |
| Content Type | Text |
| Resource Type | Article |
