# Tagged Questions

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### Regularity of Patterson-Sullivan Length function

Let $(M,g)$ be a negatively curved, closed Riemannian manifold. I'll ask the question first, then explain the involved players. This data defines the Patterson-Sullivan length function, ...
Let $I^n:=[0,1]^n$ and $T$ be a homeomorphism on $I^n$. If $T$ admits a decomposition of $I^n=A\cup B\cup C$ with $A,B,C$ Lebesgue measurable and mutually disjoint such that $$T(A)=B, T(B)=A \ ... 0answers 79 views ### A argument related measurable partitions in dynamic system X is a compact metric space, and T:X\rightarrow X be a continuous map, which is finite to one. Denoted by X_{0} the set of all points x\in X, such that for all sufficiently small ... 1answer 131 views ### Entire functions with a null real escaping set Let f be a entire function (stable on \mathbb{R}), and E_{\mathbb{R}} its real escaping set :$$E_{\mathbb{R}} = \{ x \in \mathbb{R} : f^{(k)}(x) \rightarrow_{k \to \infty} \infty \} $$We put ... 1answer 260 views ### Liouville's theorem: How to get an invariant measure? Liouville's theorem states that the `natural' 2-from is preserved under the Hamiltonian flow. Apparently this leads to an invariant measure \mu as follows d\mu = \frac{d\sigma}{|| ... 1answer 122 views ### The relations between conservative part and conservativity I revised the question. In smooth ergodic theory, a diffeomorphism is said to be conservative (I), if it preserves the Lebesgue measure. So for some of us, conservativity is just short for ... 2answers 374 views ### Fixed objects of the M endofunctor on category Meas Consider the category \operatorname{Meas} of measurable spaces: its objects are sets equipped with \sigma-algebras, and its morphisms are measurable functions between spaces. As Gerald Edgar ... 2answers 249 views ### probability measures with entropy equal to nonnegative number Is it true that for a given nonnegative number, there exists a measure-theoretical entropy value (supremum of entropies of all partitions under a measure-preserving transformation) that equals this ... 1answer 612 views ### A measure theory question Here's an interesting problem one can formulate for a student. This problem arises when considering special ergodic theorems: On a finite dimensional manifold M with a Lebesgue measure \mu, does ... 0answers 208 views ### Approximation of the radon-derivative I am looking for the following statement. Let X be a topological space and let \mu, \nu be Radon-Borel measures in the same measure class i.e. the the zero-sets are identical. Denote then by ... 2answers 707 views ### Fourier transform of x2 invariant measure Let T:\mathbb{R}/\mathbb{Z}\rightarrow \mathbb{R}/\mathbb{Z} be the map defined by T(x)=2x, and suppose that \mu is a T invariant and ergodic Borel probability measure on the space, which is ... 5answers 2k views ### Proof of Krylov-Bogoliubov Theorem Where can I find a proof (in English) of the Krylov-Bogoliubov theorem, which states if X is a compact metric space and T\colon X \to X is continuous, then there is a T-invariant Borel ... 1answer 296 views ### trivial map on \sigma-algebra \mod{}0 is trivial Hi everyone! I am currently studying the basic theory of measurable actions and need the following result, which I am not able to prove myself. It is stated without a proof, so probably it should not ... 1answer 672 views ### Given a probability \mu, can we always find a transformation T s.t. \mu is T-invariant? It is true that, under some conditions, given a measure-preserving transformation T, we can always construct a T-invariant probability. I am wondering whether we can do a converse. See Parry's ... 0answers 376 views ### Example of a quasi-Bernoulli measure which is not Gibbs? Let X=\{0,1\}^{\mathbb{N}}. For simplicity I consider measures on X only. A measure \mu is quasi-Bernoulli if there is a constant C\ge 1 such that for any finite sequences i,j,$$ C^{-1} ...
Suppose $\dot{x}=f(x)$ is a dynamical system, with $x$ in $R^n$, and $f:R^n \to R^n$ sufficiently smooth (for example, Lipschitz-continuous). Assume that $x_e$ is an unstable equilibrium point of ...