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Local maximality of horseshoe set

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0

$begingroup$

Good evening to everyone!
I am trying to prove a fact about the horseshoe set , namely that it is a locally maximal set. Some authors say that if a set $H$ is equal to $H= cap_n=-infty^infty f^n(R)$, for some open set $R$,then we call $H$ locally maximal,but this is not a definition I want to use. In fact I have to prove the following : that there is an open set $U$ containing the horseshoe set $H$ such that if $Hsubset V subset U$ and $V$ is $f$-invariant, then $V=H$.
Has anybody any idea about that? Maybe use the compactness of Cantor set in some way?
Thank you !

dynamical-systems

share|cite|improve this question

asked Mar 28 at 15:39

Petros KarajanPetros Karajan

214

$endgroup$

add a comment | 

0

$begingroup$

Good evening to everyone!
I am trying to prove a fact about the horseshoe set , namely that it is a locally maximal set. Some authors say that if a set $H$ is equal to $H= cap_n=-infty^infty f^n(R)$, for some open set $R$,then we call $H$ locally maximal,but this is not a definition I want to use. In fact I have to prove the following : that there is an open set $U$ containing the horseshoe set $H$ such that if $Hsubset V subset U$ and $V$ is $f$-invariant, then $V=H$.
Has anybody any idea about that? Maybe use the compactness of Cantor set in some way?
Thank you !

dynamical-systems

share|cite|improve this question

asked Mar 28 at 15:39

Petros KarajanPetros Karajan

214

$endgroup$

add a comment | 

$begingroup$

Good evening to everyone!
I am trying to prove a fact about the horseshoe set , namely that it is a locally maximal set. Some authors say that if a set $H$ is equal to $H= cap_n=-infty^infty f^n(R)$, for some open set $R$,then we call $H$ locally maximal,but this is not a definition I want to use. In fact I have to prove the following : that there is an open set $U$ containing the horseshoe set $H$ such that if $Hsubset V subset U$ and $V$ is $f$-invariant, then $V=H$.
Has anybody any idea about that? Maybe use the compactness of Cantor set in some way?
Thank you !

dynamical-systems

share|cite|improve this question

asked Mar 28 at 15:39

Petros KarajanPetros Karajan

214

$endgroup$

share|cite|improve this question

asked Mar 28 at 15:39

Petros KarajanPetros Karajan

214

share|cite|improve this question

asked Mar 28 at 15:39

Petros KarajanPetros Karajan

214

asked Mar 28 at 15:39

Petros KarajanPetros Karajan

214

asked Mar 28 at 15:39

Petros KarajanPetros Karajan

214