# Questions tagged [thermodynamic-formalism]

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

10 questions
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### How to estimate the pressure?

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 $G$ (...
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### 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 $T(I_i)=[0,1]$. ...
Consider the classical Multifractal Analysis, and the decomposition of the state space $X$ into level sets $$X=\bigcup_{\alpha}\left\{x\mid d_\mu(x)=\alpha\right\}\cup\left\{x\mid d_\mu(x) \,\mathrm{... 0answers 184 views ### Using topological pressure to determine a subshift of finite type I am interested in recognising graphs (or matrices, or subshifts of finite type) using topological pressure. Suppose that we play the following game: {\bf Step 1:} I write down an irreducible n x n ... 2answers 328 views ### Can a smooth diffeomorphisms of a Riemannian manifold have only positive Lyapunov exponents? Let \mu be some ergodic measure of our compact Riemannian manifold M, which is preserved by f\in Diff^{1+\beta}(M). Is it possible that all the Lyapunov exponents of \mu will be positive? ... 3answers 324 views ### Maximizing entropy under constraints This question is about an extension of the variational principle in thermodynamical formalism when one adds linear constraints to the measures. Consider the one-sided shift \sigma:\mathcal{A}^\... 1answer 164 views ### Equation of state for hard rods Some context: For ideal gases, the thermodynamic equation of state is the well-known:$$ pV = nRT \tag{1} $$where n is the amount of substance, R the universal gas constant and P,V,T are ... 2answers 714 views ### Does equality of Hodge star and symplectic star imply Kähler structure? Question The question asked is: On a manifold M equipped with a Riemann metric g and a symplectic structure \omega, with \ast the Hodge star and \ast_s the symplectic star, does \ast=\... 0answers 117 views ### Is \text{Bow}(X,T) a Banach Space? Let X=\{0,1\}^{\mathbb{N}} be the sequence space and T:X\to X the left shift mapping. Define the vector space \text{Bow}(X,T) as$$ \text{Bow}(X,T)=\{f\in C^{0}(X);~\sup_{n\in \mathbb{N}}\sup_{...
The following appears in the paper "Continuity properties of entropy" by Newhouse from 1989: Let $M$ be some smooth Riemannian compact manifold (you may assume boundary-less), and let \$f\in Diff^{1+\...