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Main areas are dynamics of automorphisms (for example, Henon maps), dynamics of endomorphisms, dynamics of foliations, and local dynamics. Eric Bedford, Tien-Cuong Dinh, John Fornaess, Misha Lyubich, Nessim Sibony, John Smiley, and their students and collaborators. One-dimensional holomorphic dynamics, functions of several complex variables (especially ...


6

If all the particles remained in a bounded domain, the virial theorem would apply. In the case of a radial inverse square power law, it states that twice the asymptotic time average of kinetic energy of the system equals minus the asymptotic time average of its potential energy. However, while the kinetic energy is always nonnegative, the potential energy ...


5

The answer is yes: there is a topologically transitive dynamical system without dense orbits. Indeed, let X be a topological space that is not separable. Let $\ K=X^{\Bbb Z},\ $ and let $\ G\ $ be the group of homeomorphism of $\ K,\ $ induced by shifts $\ s_n\ (n\in\Bbb Z)\ $ of $\ \Bbb Z:\ $ $$ \forall_{n\in\Bbb Z}\forall_{x\in\Bbb Z}\quad ...


4

The unique measure of maximal entropy $\mu_f$ supported on the Julia set of a rational map $f$ of degree $d \geq 2$ is indeed the unique balanced measure for $f$, i.e., the only probability measure $\mu$ not charging the exceptional set and satisfying $f^*\mu =d \cdot \mu$. As you already noticed in the comments, uniqueness of a measure with this property ...


4

An answer before the numerics start: First, note that for $w=u+iv \in \mathbb{C}$ and a fixed $a \in \mathbb{R}$ we have $|w+a|=\sqrt{u^2+v^2+2au+a^2}=\sqrt{|w|^2 + 2au+a^2}$ . Next, consider the image under the complex exponential of the circle $|z|=r>0$. Using polar coordinates, we get $|\exp (r\cos t+i r\sin t)|=e^{r\cos t}$. Thus to get $M_a(r)$ we ...


4

A partial solution: As mentioned by @MargaretFriedland, the desired $M_a(r)$ is the absolute maximum of $g(t):=\sqrt{{\rm{e}}^{2r\cos t}+2a{\rm{e}}^{r\cos t}\cos(r\sin t)+a^2}$. Notice that this is an even function, so it suffices to maximize over $[0,\pi]$. If $t\in\left[\frac{\pi}{2},\pi\right]$, then $\cos t<0$. So ${\rm{e}}^{2r\cos t}\in [0,1]$ and, ...


4

See Theorem 9.20 of "Topological Dynamics" by Gottschalk and Hedlund. It states that, for systems $(X,G)$ whose phase space is non-empty complete separable metric, point transitivity (a point having a dense orbit) and topological transitivity (every non-empty open set having dense orbit) are equivalent.


4

It turns out that this problem is independent of ZFC because of the following simple Theorem. Under $\mathfrak t=\mathfrak c$, every topologically transitive continuous action of a group $G$ on $\omega^*$ has a dense orbit. Proof. Let $(A_\alpha)_{\alpha\in\mathfrak c}$ be an enumeration of all infinite subsets of $\omega$. By transfinite induction we ...


3

This is a fun pair of exercises (the first one you mention is 3.2.9 and the second is 2.3.6a)! For 2.3.6a, recode to a $1$-block code $\phi$ on an irreducible edge shift $X$, suppose that $x, x' \in X$ are such that $x \neq x'$ but $\phi(x) = \phi(x')$. Consider two cases: either $x_j \neq x_j'$ for either all $j > i$ or all $j < i$; or there exist $j &...


3

Let $B$ be the smallest ball such that all $N$ particles remain inside $B$ for all $t\geq0$. Either the trajectory of one of the particles intersects $\partial B$ at some finite time $t_0$, or there is one particle and a sequence $(t_n)_{n\in\mathbb{N}}$ with $\lim \limits_{n \to \infty} t_n ~=~\infty$ such that the particle position at $t_n$ has distance $&...


3

In general $F$ is not ergodic. A very simple example can be constructed as follows: let $X=\mathbb{Z}_3=\{0,1,2\}$ and $\mu =1/3(\delta_0+\delta_1+\delta_2)$ and $T(x):=x+1$. This is an ergodic system. Let us define $Y:=\{0,1\}$ and $R\equiv 3$. Since $F=T^3=id$, it is not ergodic.


2

Begin by thinking about the dual automorphism to a hyperbolic toral automorphism. You can project the dual group onto an "expansive" direction. Any relation of the sort you are looking at can be shifted by the automorphism so that all powers are negative except for one large positive one. In the projection, this would mean that a huge number is the sum of ...


2

The measure of maximal entropy is the unique measure that is "fully invariant" in your sense. I believe that this already follows from the original proofs - indeed, it is well-known that if you take a point mass at some non-exceptional point, and keep pulling back, you will converge to the measure of maximal entropy. This should be enough to deduce the claim....


2

One should always keep in mind that the natural substrate of ergodic theory is Lebesgue-Rokhlin (aka Lebesgue or standard) measure spaces which enjoy a lot of properties not necessarily present in general measure spaces. One of these properties is an explicit description of homomorphisms of such spaces obtained by Rokhlin in his 1949 paper, which, in ...


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Assuming that a closed formula is not accessible, possibly maxmod, which calculates the maximum modulus of a complex polynomial on the unit disk, would speed a numerical attack. (Polynomial order of a few hundred is not a problem.)


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The way that the paper by Freire-Lopes-Mañé makes sense of $f^*\mu=d\mu$ is the following: ''For any Borel subset $A$ of $\Bbb{C}_\infty$ with $f\restriction_A$ injective, one has $\mu(f(A))=d.\mu(A)$.'' (See p. 46 of this paper.) One observation is that such an ergodic measure $\mu$ is either supported on the Julia set or is one of those measures with ...


1

Now I figure out that the answer of my question is false. In the paper ``Topological mixing and uniquely ergodic systems", Lehrer proved any measure-preserving system has a topologcally mixing strictly ergodic topological model, which implies that there exists a topologcally mixing strictly ergodic system $(X,T)$ such that with the unique invariant measure $\...


1

Deciding whether a set of tiles admits a periodic tiling or no tiling at all is undecidable as well. This has been shown in Y.S. Gurevich, I.I. Koryakov, Remarks on Berger's paper on the domino problem, J Sib Math J 13, 319–321 (1972). The results can also be found in the book The Classical Decision Problem by Egon Börger, Erich Grädel, Yuri Gurevich, ...


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