# Questions tagged [thermodynamic-formalism]

Thermodynamic formalism is the study of equilibrium states, Gibbs measures and topological pressure for dynamical systems.

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### Can a smooth diffeomorphisms of a Riemannian manifold have only positive Lyapunov exponents?

Let $\mu$ be some ergodic measure of our compact Riemannian manifold $M$, which is preserved by $f\in Diff^{1+\beta}(M)$. Is it possible that all the Lyapunov exponents of $\mu$ will be positive? ...
This question is about an extension of the variational principle in thermodynamical formalism when one adds linear constraints to the measures. Consider the one-sided shift $\sigma:\mathcal{A}^\... 1answer 196 views ### Estimates of Hausdorff dimension (and its derivatives) For example, the cookie cutter maps, say$T:I_1 \cup I_2 \subset [0,1] \to [0,1] $is a$C^2$map such that$|T'|>1$and provided$I_1$and$I_2$are disjoint closed intervals and$T(I_i)=[0,1]$. ... 0answers 118 views ### Is$\text{Bow}(X,T)$a Banach Space? Let$X=\{0,1\}^{\mathbb{N}}$be the sequence space and$T:X\to X$the left shift mapping. Define the vector space$\text{Bow}(X,T)$as $$\text{Bow}(X,T)=\{f\in C^{0}(X);~\sup_{n\in \mathbb{N}}\sup_{... 0answers 202 views ### Using topological pressure to determine a subshift of finite type I am interested in recognising graphs (or matrices, or subshifts of finite type) using topological pressure. Suppose that we play the following game: {\bf Step 1:} I write down an irreducible n x n ... 1answer 160 views ### Multifractal Analysis and Dimension Spectrum of Unions Consider the classical Multifractal Analysis, and the decomposition of the state space X into level sets$$X=\bigcup_{\alpha}\left\{x\mid d_\mu(x)=\alpha\right\}\cup\left\{x\mid d_\mu(x) \,\mathrm{... 2answers 810 views ### Does equality of Hodge star and symplectic star imply Kähler structure? Question The question asked is: On a manifold$M$equipped with a Riemann metric$g$and a symplectic structure$\omega$, with$\ast$the Hodge star and$\ast_s$the symplectic star, does$\ast=\...
I have a finite collection of diffeomorphisms $g_1,\cdots,g_n$ taking the unit interval $I$ to disjoint subintervals $I_1, I_2,\cdots,I_n$. If $G$ is the semigroup they generate, the limit set of $G$ (...