Dimension of a self-similar measure is never greater than its similarity dimension

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Assume $mu = sum_i in Lambda p_i cdot mu circ phi_i^-1$ is a self-similar measure on $mathbbR$ associated to an iterated function system $Phi = lbrace phi_i rbrace_i in Lambda$, $phi_i(x) = r_i x + a_i$. I want to prove the following inequality;
$$dim mu leq min lbrace 1, texts-dim mu rbrace,$$
where
$$
texts-dim mu = dfracsum_i in Lambda p_i log p_isum_i in Lambda p_i log r_i
$$
is the similarity dimension of $mu$. According to the author of the paper I am reading, this inequality is trivial. An identical one exists for the dimension of the attractor of $Phi$ and that is quite easy to prove.



The $texts-dim mu leq 1$ part is trivial, $texts-dim mu leq dim X leq 1$ since $mu$ is supported on the attractor $X$ but what about $dim mu leq texts-dim mu$?




measure-theory fractals dimension-theory




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  • 1




    What paper are you reading?
    – Xander Henderson
    Aug 1 at 22:17














up vote
0
down vote

favorite












Assume $mu = sum_i in Lambda p_i cdot mu circ phi_i^-1$ is a self-similar measure on $mathbbR$ associated to an iterated function system $Phi = lbrace phi_i rbrace_i in Lambda$, $phi_i(x) = r_i x + a_i$. I want to prove the following inequality;
$$dim mu leq min lbrace 1, texts-dim mu rbrace,$$
where
$$
texts-dim mu = dfracsum_i in Lambda p_i log p_isum_i in Lambda p_i log r_i
$$
is the similarity dimension of $mu$. According to the author of the paper I am reading, this inequality is trivial. An identical one exists for the dimension of the attractor of $Phi$ and that is quite easy to prove.



The $texts-dim mu leq 1$ part is trivial, $texts-dim mu leq dim X leq 1$ since $mu$ is supported on the attractor $X$ but what about $dim mu leq texts-dim mu$?




measure-theory fractals dimension-theory




share|cite|improve this question















  • 1




    What paper are you reading?
    – Xander Henderson
    Aug 1 at 22:17












up vote
0
down vote

favorite









up vote
0
down vote

favorite











Assume $mu = sum_i in Lambda p_i cdot mu circ phi_i^-1$ is a self-similar measure on $mathbbR$ associated to an iterated function system $Phi = lbrace phi_i rbrace_i in Lambda$, $phi_i(x) = r_i x + a_i$. I want to prove the following inequality;
$$dim mu leq min lbrace 1, texts-dim mu rbrace,$$
where
$$
texts-dim mu = dfracsum_i in Lambda p_i log p_isum_i in Lambda p_i log r_i
$$
is the similarity dimension of $mu$. According to the author of the paper I am reading, this inequality is trivial. An identical one exists for the dimension of the attractor of $Phi$ and that is quite easy to prove.



The $texts-dim mu leq 1$ part is trivial, $texts-dim mu leq dim X leq 1$ since $mu$ is supported on the attractor $X$ but what about $dim mu leq texts-dim mu$?




measure-theory fractals dimension-theory




share|cite|improve this question











Assume $mu = sum_i in Lambda p_i cdot mu circ phi_i^-1$ is a self-similar measure on $mathbbR$ associated to an iterated function system $Phi = lbrace phi_i rbrace_i in Lambda$, $phi_i(x) = r_i x + a_i$. I want to prove the following inequality;
$$dim mu leq min lbrace 1, texts-dim mu rbrace,$$
where
$$
texts-dim mu = dfracsum_i in Lambda p_i log p_isum_i in Lambda p_i log r_i
$$
is the similarity dimension of $mu$. According to the author of the paper I am reading, this inequality is trivial. An identical one exists for the dimension of the attractor of $Phi$ and that is quite easy to prove.



The $texts-dim mu leq 1$ part is trivial, $texts-dim mu leq dim X leq 1$ since $mu$ is supported on the attractor $X$ but what about $dim mu leq texts-dim mu$?





measure-theory fractals dimension-theory





share|cite|improve this question










share|cite|improve this question




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asked Aug 1 at 9:08









Uuno

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  • 1




    What paper are you reading?
    – Xander Henderson
    Aug 1 at 22:17












  • 1




    What paper are you reading?
    – Xander Henderson
    Aug 1 at 22:17







1




1




What paper are you reading?
– Xander Henderson
Aug 1 at 22:17




What paper are you reading?
– Xander Henderson
Aug 1 at 22:17















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