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Let $X$ be a compact metric space, and $\mu$ a probability measure on $X$ with $\text{supp} \ \mu = X$. Suppose $T: X \to X$ is continuous, measure preserving and uniformly transitive in the sense that it satisfies the following two conditions:

i) The orbit of every point is dense in $X$.

ii) For every $\varepsilon > 0$, there exists some $\delta > 0$ such that for every $x \in X$, and $r > 0$ with $\mu(B_r (x)) > \varepsilon$, we have $T(B_s (T^{-1} (x)) \subset B_r (x)$ for all $ s > 0$ such that $\mu(B_s (T^{-1} (x)) < \delta$.

Question: Does it follow that $T$ is ergodic?

Let $X$ be a compact metric space, and $\mu$ a probability measure on $X$ with $\text{supp} \ \mu = X$. Suppose $T: X \to X$ is continuous, and uniformly transitive in the sense that it satisfies the following two conditions:

i) The orbit of every point is dense in $X$.

ii) For every $\varepsilon > 0$, there exists some $\delta > 0$ such that for every $x \in X$, and $r > 0$ with $\mu(B_r (x)) > \varepsilon$, we have $T(B_s (T^{-1} (x)) \subset B_r (x)$ for all $ s > 0$ such that $\mu(B_s (T^{-1} (x)) < \delta$.

Question: Does it follow that $T$ is ergodic?

Let $X$ be a compact metric space, and $\mu$ a probability measure on $X$ with $\text{supp} \ \mu = X$. Suppose $T: X \to X$ is continuous, measure preserving and uniformly transitive in the sense that it satisfies the following two conditions:

i) The orbit of every point is dense in $X$.

ii) For every $\varepsilon > 0$, there exists some $\delta > 0$ such that for every $x \in X$, and $r > 0$ with $\mu(B_r (x)) > \varepsilon$, we have $T(B_s (T^{-1} (x)) \subset B_r (x)$ for all $ s > 0$ such that $\mu(B_s (T^{-1} (x)) < \delta$.

Question: Does it follow that $T$ is ergodic?

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Nate River
Nate River
  • 6.3k
  • 2
  • 23
  • 99

Let $X$ be a compact metric space, and $\mu$ a probability measure on $X$ with $\text{supp} \ \mu = X$. Suppose $T: X \to X$ is continuous, and uniformly transitive in the sense that it satisfies the following two conditions:

i) The orbit of every point is dense in $X$.

ii) For every $\varepsilon > 0$, there exists some $\delta > 0$ such that for every $x \in X$, and $r > 0$ with $\mu(B_r (x)) < \varepsilon$$\mu(B_r (x)) > \varepsilon$, we have $T(B_s (T^{-1} (x)) \subset B_r (x)$ for all $ s > 0$ such that $\mu(B_s (T^{-1} (x)) < \delta$.

Question: Does it follow that $T$ is ergodic?

Let $X$ be a compact metric space, and $\mu$ a probability measure on $X$ with $\text{supp} \ \mu = X$. Suppose $T: X \to X$ is continuous, and uniformly transitive in the sense that it satisfies the following two conditions:

i) The orbit of every point is dense in $X$.

ii) For every $\varepsilon > 0$, there exists some $\delta > 0$ such that for every $x \in X$, and $r > 0$ with $\mu(B_r (x)) < \varepsilon$, we have $T(B_s (T^{-1} (x)) \subset B_r (x)$ for all $ s > 0$ such that $\mu(B_s (T^{-1} (x)) < \delta$.

Question: Does it follow that $T$ is ergodic?

Let $X$ be a compact metric space, and $\mu$ a probability measure on $X$ with $\text{supp} \ \mu = X$. Suppose $T: X \to X$ is continuous, and uniformly transitive in the sense that it satisfies the following two conditions:

i) The orbit of every point is dense in $X$.

ii) For every $\varepsilon > 0$, there exists some $\delta > 0$ such that for every $x \in X$, and $r > 0$ with $\mu(B_r (x)) > \varepsilon$, we have $T(B_s (T^{-1} (x)) \subset B_r (x)$ for all $ s > 0$ such that $\mu(B_s (T^{-1} (x)) < \delta$.

Question: Does it follow that $T$ is ergodic?

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Nate River
Nate River
  • 6.3k
  • 2
  • 23
  • 99

Is a “uniformly minimal” dynamical system ergodic?

Let $X$ be a compact metric space, and $\mu$ a probability measure on $X$ with $\text{supp} \ \mu = X$. Suppose $T: X \to X$ is continuous, and uniformly transitive in the sense that it satisfies the following two conditions:

i) The orbit of every point is dense in $X$.

ii) For every $\varepsilon > 0$, there exists some $\delta > 0$ such that for every $x \in X$, and $r > 0$ with $\mu(B_r (x)) < \varepsilon$, we have $T(B_s (T^{-1} (x)) \subset B_r (x)$ for all $ s > 0$ such that $\mu(B_s (T^{-1} (x)) < \delta$.

Question: Does it follow that $T$ is ergodic?