First let me explain problem in general case

If $T:\mathbb R^n\to \mathbb R^n$ and $S:\mathbb R^n\to \mathbb R^n$ are two conjugated linear Dynamical Systems which means there exist homeomorphism $h:\mathbb R^n\to \mathbb R^n$ such that $ h(S(x))=T(h(x))$

Now we know $h$ induced another metric $d_h$ on $\mathbb R^n$ which is topological equivalent to euclidean metric now we raise a question. no we can compute metric entropy via two metrics

1. $H_d(T)=?$

2. $H_{d_{h}}(T)=?$

by pesin formula we know that $$H_d(T)=\sum_{\mu\in Spec(T)}\log{|\mu|}$$

How compute metric entropy with new metric ??

W.L.O.G we can simplify this problem and take$n=1$ and $$T(x)=2x$$ $$S(x)=3x$$ are two linear systems that are conjugate .

we can contract question in following form Compute entropy of $T(x)=2x$ by following metric??

$$d_1(x,y)=|x-y|$$ $$d_2(x,y)=|logx-logy|$$

$$d_3(x,y)=|x^c-y^c|$$

I started to compute generating set or separated set but working with these was so though for i thought should be some short cut

Thanks for any hints

First let me explain problem in general case

If $T:\mathbb R^n\to \mathbb R^n$ and $S:\mathbb R^n\to \mathbb R^n$ are two conjugated linear Dynamical Systems which means there exist homeomorphism $h:\mathbb R^n\to \mathbb R^n$ such that $ h(S(x))=T(h(x))$

Now we know $h$ induced another metric $d_h$ on $\mathbb R^n$ which is topological equivalent to euclidean metric now we raise a question. no we can compute metric entropy via two metrics

1. $H_d(T)=?$

2. $H_{d_{h}}(T)=?$

by pesin formula we know that $$H_d(T)=\sum_{{\mu\in \{Spec(T)\cap{|\mu|\gt1}}\}} log{|\mu|}$$

How compute metric entropy with new metric ??

W.L.O.G we can simplify this problem and take$n=1$ and $$T(x)=2x$$ $$S(x)=3x$$ are two linear systems that are conjugate .

we can contract question in following form Compute entropy of $T(x)=2x$ by following metric??

$$d_1(x,y)=|x-y|$$ $$d_2(x,y)=|logx-logy|$$

$$d_3(x,y)=|x^c-y^c|$$

I started to compute generating set or separated set but working with these was so though for i thought should be some short cut

Thanks for any hints

First let me explain problem in general case

If $T:\mathbb R^n\to \mathbb R^n$ and $S:\mathbb R^n\to \mathbb R^n$ are two conjugated linear Dynamical Systems which means there exist homeomorphism $h:\mathbb R^n\to \mathbb R^n$ such that $ h(S(x))=T(h(x))$

Now we know $h$ induced another metric $d_h$ on $\mathbb R^n$ which is topological equivalent to euclidean metric now we raise a question. no we can compute metric entropy via two metrics

1. $H_d(T)=?$

2. $H_{d_{h}}(T)=?$

by pesin formula we know that $$H_d(T)=\sum_{\mu\in Spec(T)}\log{|\mu|}$$

How compute metric entropy with new metric ??

W.L.O.G we can simplify this problem and take$n=1$ and $$T(x)=2x$$ $$S(x)=3x$$ are two linear systems that are conjugate .

we can contract question in following form Compute entropy of $T(x)=2x$ by following metric??

$$d_1(x,y)=|x-y|$$ $$d_2(x,y)=|logx-logy|$$

$$d_3(x,y)=|x^c-y^c|$$ THanks for any hints

First let me explain problem in general case

If $T:\mathbb R^n\to \mathbb R^n$ and $S:\mathbb R^n\to \mathbb R^n$ are two conjugated linear Dynamical Systems which means there exist homeomorphism $h:\mathbb R^n\to \mathbb R^n$ such that $ h(S(x))=T(h(x))$

Now we know $h$ induced another metric $d_h$ on $\mathbb R^n$ which is topological equivalent to euclidean metric now we raise a question. no we can compute metric entropy via two metrics

1. $H_d(T)=?$

2. $H_{d_{h}}(T)=?$

by pesin formula we know that $$H_d(T)=\sum_{\mu\in Spec(T)}\log{|\mu|}$$

How compute metric entropy with new metric ??

W.L.O.G we can simplify this problem and take$n=1$ and $$T(x)=2x$$ $$S(x)=3x$$ are two linear systems that are conjugate .

we can contract question in following form Compute entropy of $T(x)=2x$ by following metric??

$$d_1(x,y)=|x-y|$$ $$d_2(x,y)=|logx-logy|$$

$$d_3(x,y)=|x^c-y^c|$$

I started to compute generating set or separated set but working with these was so though for i thought should be some short cut

Thanks for any hints