Skip to main content
Became Hot Network Question
added 78 characters in body
Source Link

Let $\pi\colon(X,T)\to (Y,T)$ be a factor map between minimal subshifts. Suppose there exists $\tilde{Y} \subseteq Y$ such that

  1. $\# \pi^{-1}(y) = 1$ for all $y \in \tilde{Y}$.
  2. $\tilde{Y}$ is a residual subset of $Y$ i.e. $\overline{Y\setminus \tilde{Y}}$ has empty interior$\tilde{Y} = \bigcap_{n=1}^\infty\tilde{Y}_n$ for some collection $\{\tilde{Y}_1,\tilde{Y}_2,\dots\}$ of dense open subsets of $Y$.
  3. $\mu(\tilde{Y}) = 1$ for every $T$-invariant probability measure in $Y$.

Is it true that $\pi$ is injective?

Let $\pi\colon(X,T)\to (Y,T)$ be a factor map between minimal subshifts. Suppose there exists $\tilde{Y} \subseteq Y$ such that

  1. $\# \pi^{-1}(y) = 1$ for all $y \in \tilde{Y}$.
  2. $\tilde{Y}$ is a residual subset of $Y$ i.e. $\overline{Y\setminus \tilde{Y}}$ has empty interior.
  3. $\mu(\tilde{Y}) = 1$ for every $T$-invariant probability measure in $Y$.

Is it true that $\pi$ is injective?

Let $\pi\colon(X,T)\to (Y,T)$ be a factor map between minimal subshifts. Suppose there exists $\tilde{Y} \subseteq Y$ such that

  1. $\# \pi^{-1}(y) = 1$ for all $y \in \tilde{Y}$.
  2. $\tilde{Y}$ is a residual subset of $Y$ i.e. $\tilde{Y} = \bigcap_{n=1}^\infty\tilde{Y}_n$ for some collection $\{\tilde{Y}_1,\tilde{Y}_2,\dots\}$ of dense open subsets of $Y$.
  3. $\mu(\tilde{Y}) = 1$ for every $T$-invariant probability measure in $Y$.

Is it true that $\pi$ is injective?

fixed typos
Source Link
YCor
  • 63.9k
  • 5
  • 187
  • 286

Does this strong form of being almost 1-to-1 implie inyectivityimply injectivity?

Let $\pi\colon(X,T)\to (Y,T)$ be a factor map between minimal subshifts. Suppose there exists $\tilde{Y} \subseteq Y$ such that

  1. $\# \pi^{-1}(y) = 1$ for all $y \in \tilde{Y}$.
  2. $\tilde{Y}$ is a residual subset of $Y$ i.e. $\overline{Y\setminus \tilde{Y}}$ has empty interior.
  3. $\mu(\tilde{Y}) = 1$ for allevery $T$-invariant probability measure in $Y$.

Is it true that $\pi$ is a conjugacyinjective?

Does this strong form of being almost 1-to-1 implie inyectivity?

Let $\pi\colon(X,T)\to (Y,T)$ be a factor map between minimal subshifts. Suppose there exists $\tilde{Y} \subseteq Y$ such that

  1. $\# \pi^{-1}(y) = 1$ for all $y \in \tilde{Y}$.
  2. $\tilde{Y}$ is a residual subset of $Y$ i.e. $\overline{Y\setminus \tilde{Y}}$ has empty interior.
  3. $\mu(\tilde{Y}) = 1$ for all $T$-invariant probability measure in $Y$.

Is it true that $\pi$ is a conjugacy?

Does this strong form of being almost 1-to-1 imply injectivity?

Let $\pi\colon(X,T)\to (Y,T)$ be a factor map between minimal subshifts. Suppose there exists $\tilde{Y} \subseteq Y$ such that

  1. $\# \pi^{-1}(y) = 1$ for all $y \in \tilde{Y}$.
  2. $\tilde{Y}$ is a residual subset of $Y$ i.e. $\overline{Y\setminus \tilde{Y}}$ has empty interior.
  3. $\mu(\tilde{Y}) = 1$ for every $T$-invariant probability measure in $Y$.

Is it true that $\pi$ is injective?

Source Link

Does this strong form of being almost 1-to-1 implie inyectivity?

Let $\pi\colon(X,T)\to (Y,T)$ be a factor map between minimal subshifts. Suppose there exists $\tilde{Y} \subseteq Y$ such that

  1. $\# \pi^{-1}(y) = 1$ for all $y \in \tilde{Y}$.
  2. $\tilde{Y}$ is a residual subset of $Y$ i.e. $\overline{Y\setminus \tilde{Y}}$ has empty interior.
  3. $\mu(\tilde{Y}) = 1$ for all $T$-invariant probability measure in $Y$.

Is it true that $\pi$ is a conjugacy?