Ian Morris
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Registered User
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I am a lecturer in mathematics at the University of Surrey in Guildford, England, where I have been working since January 2012. I live in London.
In general I am interested in applications of ergodic theory to other areas of mathematics. Some particular problems which interest me at the moment include the analysis of algorithms using Ruelle operators, the topological dynamics of control systems, and the joint spectral characteristics of sets of linear operators. |
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1d |
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Sequences equidistributed modulo 1 added 310 characters in body |
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1d |
revised |
Sequences equidistributed modulo 1 added 253 characters in body; added 19 characters in body |
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1d |
answered | Sequences equidistributed modulo 1 |
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2d |
revised |
Blue and red balls puzzle added 2 characters in body |
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2d |
revised |
Blue and red balls puzzle edited tags |
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2d |
revised |
Blue and red balls puzzle added 3 characters in body; added 3 characters in body |
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2d |
answered | Blue and red balls puzzle |
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May 10 |
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Variational Principle for the Entropy It's Theorem 8.6 in Walters. |
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Apr 18 |
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Recurrence and transience of cocycle over a dynamical system Perhaps I am missing something, but how is the inequality $\lim_{n \to \infty} \int \phi_n d\mu \geq \int \liminf_{n \to \infty}\phi_n d\mu$ justified? Fatou's lemma does not work here, for example, because the functions $\phi_n$ might fail to be uniformly bounded below. |
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Apr 17 |
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Recurrence and transience of cocycle over a dynamical system added 210 characters in body; added 7 characters in body |
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Apr 17 |
answered | Recurrence and transience of cocycle over a dynamical system |
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Apr 17 |
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Recurrence and transience of cocycle over a dynamical system What is the question? |
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Mar 31 |
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Iterates converging to a continuous map edited tags |
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Mar 22 |
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Liverani’s CLT (a question) The absolute value of the weak limit of $f_n$ does not have to equal the weak limit of the sequence of absolute values $|f_n|$. For example, in $L^2([0,1])$ take $f_n(x)=sin(nx)$ to obtain $f_n \to 0$ weakly and $|f_n| \to \frac{1}{2}$ weakly. |
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Mar 22 |
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Liverani’s CLT (a question) Let $(f_n)$ be a sequence which converges in the weak topology of $L^2$ to $f$, and converges a.e. to $g$. To complete the proof we must show that $f=g$. Let $\delta>0$ and choose a set $E$ with $m(E)>1-\delta$ such that $f$ is bounded on $E$. Since $f_n \to g$ a.e, by Egoroff's Thm we can find $F \subset E$ with $m(F)>1-2\delta$ such that $f_n \to g$ uniformly on $F$. By weak convergence $\int f_n(g-f)\chi_F dm \to \int f(g-f)\chi_F dm$, and by uniform convergence on $F$ also $\int f_n(g-f)\chi_F dm \to \int g(g-f)\chi_F dm$, so $\int_F(g-f)^2=0$ and $f=g$ except on a set of measre $2\delta$. |
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Mar 22 |
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Liverani’s CLT (a question) If $(e_n)$ is an orthonormal basis sequence for $L^2$ then its weak limit is zero but it does not converge in the $L^2$ distance. Your argument shows -- correctly I think, if you directly use the definition of weak convergence to justify $E(D_1(\lambda_n I_A)) \to E(D_1I_A)$ instead of attempting to use norm convergence in $L^2$ and $L^1$ -- that $D(\lambda_n)$ has a limit in the weak topology, and that the conditional expectation with respect to $\mathcal{F}_1$ of that limit is zero. It remains only to show that the weak limit really is $D_1$: math.stackexchange.com/questions/160306 |
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Mar 22 |
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Liverani’s CLT (a question) I am unsure about the details of your application of Alaoglu's theorem: in general a bounded sequence in $L^2$ will not have a convergent subsequence with respect to the $L^2$ distance, only with respect to the weak topology. However, $g \mapsto \mathbb{E}(g\chi_A)$ is a continuous linear functional on $L^2$ so Alaoglu's theorem delivers the result you need and $\mathbb{E}(D_1(\lambda_n)\chi_A)$ does converge to $\mathbb{E}(f\chi_A)$ where $f$ is the limit of the subsequence. I don't recall the details, but it should not be hard to show that the weak limit and a.e. limit agree when both exist. |
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Mar 22 |
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Silly question about mixing Here is a correct modification of Lemma 2.4: if $\mu(T^{-n}A \cap B)$ is eventually nonzero whenever $\mu(A)$ and $\mu(B)$ are both nonzero then $T$ is light mixing. Proof: suppose that $T$ is not light mixing. Choose $A,B$ with $\mu(A),\mu(B)>0$ and $\liminf_{n \to \infty} \mu(T^{-n}\cap B)=0$. Choose a strictly increasing sequence $(n_k)$ such that $\mu(T^{-n_k}A \cap B)<3^{-k}\mu(B)$ for all $k \geq 1$. Let $C:=B \setminus \bigcup_{k=1}^\infty \left(T^{-n_k}A \cap B\right)$. Then $\mu(C)>\frac{1}{2}\mu(B)>0$ and $\mu(T^{-n_k}A \cap C)=0$ for all $k$, a contradiction. |
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Mar 22 |
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Silly question about mixing You're right: if $T$ is not lightly mixing then there is no reason why we should be able to find $E$ such that $\liminf_{n \to \infty}\mu(T^{-n}E\cap E)=0$. I think that the authors err when they state that in order to check light mixing it is sufficient to check the case $A=B$: this is fine for weak, strong and probably mild mixing because in those cases the relevant expressions are linear in $\chi_A$ and $\chi_B$, but lim inf is of course not linear. |
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Mar 22 |
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Silly question about mixing What a fascinating paper! If I'm not wrong, when $T$ is invertible, Lemma 2.4 in that article shows that Étienne's condition (with positive-measure rather than nonempty intersections) is precisely light mixing. |
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Mar 22 |
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Silly question about mixing Perhaps you want $\mu(T^{-n}A \cap B)>0$ rather than nonemptiness, since the former is more natural in a probability space. This is certainly not a silly question. In the positive-measure form this condition implies weak mixing by Theorem 4.31 in Furstenberg's book "Recurrence in Ergodic Theory and Combinatorial Number Theory". In Parry's book "Topics in Ergodic Theory" (p.89) a transformation is discussed which is weak mixing but does not meet this condition. I suspect that the answer to your question is positive but it may not be widely known. |
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Mar 20 |
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Open problems in PDEs, dynamical systems, mathematical physics deleted 13 characters in body |
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Mar 20 |
answered | Open problems in PDEs, dynamical systems, mathematical physics |
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Mar 19 |
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Liverani’s CLT (a question) added 264 characters in body |
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Mar 19 |
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Liverani’s CLT (a question) added 6 characters in body |
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Mar 19 |
answered | Liverani’s CLT (a question) |
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Mar 16 |
answered | Equivalence of two definitions of Lyapunov exponents |
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Mar 13 |
awarded | ● Enlightened |
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Mar 13 |
accepted | Non-existence of ergodic measures |
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Mar 10 |
awarded | ● Nice Answer |
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Mar 9 |
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Non-existence of ergodic measures added 90 characters in body |
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Mar 9 |
answered | Non-existence of ergodic measures |
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Mar 6 |
awarded | ● Nice Answer |
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Feb 25 |
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Characterising ergodicity of continuous maps edited tags |
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Feb 24 |
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Characterising ergodicity of continuous maps @Julian: Every invariant probability measure of a minimal transformation is fully supported, because otherwise its support would be a nonempty closed invariant proper subset, contradicting minimality. So the two statements are equivalent. |
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Feb 24 |
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Characterising ergodicity of continuous maps @Julian: this is equivalent to asking for a condition on $X$ such that every minimal transformation on $X$ is uniquely ergodic, i.e. has only one invariant measure. (If a transformation has two distinct invariant measures then a strict linear combination of the two is never ergodic.) Such conditions do exist: finite spaces $X$ have this property, as does the circle (I think) but as Anthony says this is a severly restrictive requirement. The broader stroke of your question seems to be whether ergodicity can be easily characterised using only topological concepts. The answer to this is "No". |
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Feb 24 |
accepted | Characterising ergodicity of continuous maps |
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Feb 24 |
answered | Characterising ergodicity of continuous maps |
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Feb 8 |
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Invariant measures for Cellular automata If I understand you correctly then the action of this system on the closed subset $\{0,1\}^{\mathbb{N}}$ is simply the shift, so this system admits a host of shift-invariant ergodic measures supported on $\{0,1\}^{\mathbb{N}}$. Or do you want the measure to be fully supported? |
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Jan 30 |
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Metrization of weak convergence of signed measures Do you perhaps want $\Omega$ to be metric? It is not clear to me how the Prokhorov and Wasserstein metrics on $\mathcal{P}(\Omega)$ might be defined in the absence of a metric on $\Omega$, and I am suspicious of the suggestion that $\mathcal{P}(\Omega)$ is metrisable even when $\Omega$ is not (for example in the case where $\Omega$ is the Stone-Čech compactification of $\mathbb{N}$). In fact, can we not identify $\Omega$ with the subset of $\mathcal{P}(\Omega)$ which comprises the Dirac measures, and deduce that if $\mathcal{P}(\Omega)$ is metrisable then $\Omega$ must be metrisable also? |
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Dec 29 |
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Old books still used added 6 characters in body |
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Dec 29 |
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Old books still used +1 for Kato. I've had my copy for six years so far and I've learned something new from it in every year. |
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Dec 29 |
answered | Old books still used |
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Dec 28 |
accepted | Estimate entropy of a binary process in terms of decay of correlations |
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Dec 26 |
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Estimate entropy of a binary process in terms of decay of correlations Thinking back, I have absolutely no idea why I used powers of two! |
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Dec 25 |
answered | Estimate entropy of a binary process in terms of decay of correlations |
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Dec 6 |
awarded | ● Enlightened |
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Dec 6 |
awarded | ● Nice Answer |
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Dec 3 |
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Connection between properties of Dynamical and Ergodic Systems Topological transitivity is equivalent to the existence of a dense orbit. Indeed, the books by Walters and by Katok and Hasselblatt take this as the definition. My suggestions for 1 and 8 both have a dense orbit. Using the other popular version of the definition, if $U$ and $V$ are nonempty open sets then it is clear that both must contain a point from the connecting homoclinic or heteroclinic orbit (call them $x \in U$ and $y \in V$). Since by definition $T^nx=y$ for some $n \in \mathbb{Z}$ we have $y \in T^nU \cap V \neq \emptyset$. |
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Dec 1 |
awarded | ● Nice Question |
