Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userSymmetry of entropy in higher rank diagonalizable actions and measure classification
221
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
An important consequence of the theory of entropy of Z-actions is that the events measurable with respect to the far future coincide (modulo null sets) with those measurable with respect to the distant past, and that measuring the entropy using the past will give the same value as measuring it using the future. In this paper we show that for measures invariant under multiparameter algebraic actions if the entropy attached to coarse Lyapunov foliations fail to display a stronger symmetry property of a similar type this forces the measure to be invariant under non-trivial unipotent groups. Some consequences of this phenomenon are noted.
rate research
Read More
Let $Gamma$ be a lattice in ${rm SL}(n, mathbb R)$ with $ngeq 3$ and $mathcal S$ be a closed surface. Then $Gamma$ has no distal minimal action on $mathcal S$.
We show that joinings of higher rank torus actions on S-arithmetic quotients of semi-simple or perfect algebraic groups must be algebraic.
We show that, for countable sofic groups, a Bernoulli action with infinite entropy base has infinite entropy with respect to every sofic approximation sequence. This builds on the work of Lewis Bowen in the case of finite entropy base and completes the computation of measure entropy for Bernoulli actions over countable sofic groups. One consequence is that such a Bernoulli action fails to have a generating countable partition with finite entropy if the base has infinite entropy, which in the amenable case is well known and in the case that the acting group contains the free group on two generators was established by Bowen using a different argument.
Recent papers of the authors have completely described the hyperbolic actions of several families of classically studied solvable groups. A key tool for these investigations is the machinery of confining subsets of Caprace, Cornulier, Monod, and Tessera, which applies, in particular, to solvable groups with virtually cyclic abelianizations. In this paper, we extend this machinery and give a correspondence between the hyperbolic actions of certain solvable groups with higher rank abelianizations and confining subsets of these more general groups. We then apply this extension to give a complete description of the hyperbolic actions of a family of groups related to Baumslag-Solitar groups.
Let $Lambda$ be a complex manifold and let $(f_lambda)_{lambdain Lambda}$ be a holomorphic family of rational maps of degree $dgeq 2$ of $mathbb{P}^1$. We define a natural notion of entropy of bifurcation, mimicking the classical definition of entropy, by the parametric growth rate of critical orbits. We also define a notion a measure-theoretic bifurcation entropy for which we prove a variational principle: the measure of bifurcation is a measure of maximal entropy. We rely crucially on a generalization of Yomdins bound of the volume of the image of a dynamical ball. Applying our technics to complex dynamics in several variables, we notably define and compute the entropy of the trace measure of the Green currents of a holomorphic endomorphism of $mathbb{P}^k$.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
