# Ian Morris

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## Registered User

 Name Ian Morris Member for 3 years Seen 10 hours ago Website Location Guildford, United Kingdom. Age 34
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.
 1d revised Sequences equidistributed modulo 1added 310 characters in body 1d revised Sequences equidistributed modulo 1added 253 characters in body; added 19 characters in body 1d answered Sequences equidistributed modulo 1 2d revised Blue and red balls puzzleadded 2 characters in body 2d revised Blue and red balls puzzleedited tags 2d revised Blue and red balls puzzleadded 3 characters in body; added 3 characters in body 2d answered Blue and red balls puzzle May10 comment Variational Principle for the EntropyIt's Theorem 8.6 in Walters. Apr18 comment Recurrence and transience of cocycle over a dynamical systemPerhaps 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. Apr17 revised Recurrence and transience of cocycle over a dynamical systemadded 210 characters in body; added 7 characters in body Apr17 answered Recurrence and transience of cocycle over a dynamical system Apr17 comment Recurrence and transience of cocycle over a dynamical systemWhat is the question? Mar31 revised Iterates converging to a continuous mapedited tags Mar22 comment 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. Mar22 comment 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 $2\delta$. Mar22 comment 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 Mar22 comment 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. Mar22 comment Silly question about mixingHere 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. Mar22 comment Silly question about mixingYou'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. Mar22 comment Silly question about mixingWhat 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. Mar22 comment Silly question about mixingPerhaps 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. Mar20 revised Open problems in PDEs, dynamical systems, mathematical physicsdeleted 13 characters in body Mar20 answered Open problems in PDEs, dynamical systems, mathematical physics Mar19 revised Liverani’s CLT (a question)added 264 characters in body Mar19 revised Liverani’s CLT (a question)added 6 characters in body Mar19 answered Liverani’s CLT (a question) Mar16 answered Equivalence of two definitions of Lyapunov exponents Mar13 awarded ● Enlightened Mar13 accepted Non-existence of ergodic measures Mar10 awarded ● Nice Answer Mar9 revised Non-existence of ergodic measuresadded 90 characters in body Mar9 answered Non-existence of ergodic measures Mar6 awarded ● Nice Answer Feb25 revised Characterising ergodicity of continuous mapsedited tags Feb24 comment 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. Feb24 comment 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". Feb24 accepted Characterising ergodicity of continuous maps Feb24 answered Characterising ergodicity of continuous maps Feb8 comment Invariant measures for Cellular automataIf 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? Jan30 comment Metrization of weak convergence of signed measuresDo 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? Dec29 revised Old books still usedadded 6 characters in body Dec29 comment 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. Dec29 answered Old books still used Dec28 accepted Estimate entropy of a binary process in terms of decay of correlations Dec26 comment Estimate entropy of a binary process in terms of decay of correlationsThinking back, I have absolutely no idea why I used powers of two! Dec25 answered Estimate entropy of a binary process in terms of decay of correlations Dec6 awarded ● Enlightened Dec6 awarded ● Nice Answer Dec3 comment Connection between properties of Dynamical and Ergodic SystemsTopological 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 $y \in T^nU \cap V \neq \emptyset$. Dec1 awarded ● Nice Question