Questions tagged [thermodynamic-formalism]

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

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Factor map between subshifts preserving topological pressure (or measure-theoretic entropy)

Let $G$ be a countable amenable group and let $X,Y$ be subshifts with finite alphabet over $G$. Suppose that $h(X) = h(Y)$ (equal topological entropy). I am interested in continuous factor maps $\pi: ...
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81 views

Gurevich's entropy and topological entropy in a countable Markov shift

Good afternoon, I understand that Gurevich's entropy and topological entropy coincide when the countable Markov shift is topologically mixing (right?) Does anyone know of an example or a reference ...
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3answers
323 views

Introduction to information geometry and/or geometric control theory

Some background: I'vebeen searching for a research project to work through my grad studies and I found information geometry like a strong candidate but the amount of work out there is overwhelming. I ...
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0answers
53 views

Examples of non-uniqueness of the equilibrium states

Let $f:X\rightarrow X$ be an Axiom $A$ diffeomorphism on a compact metric space $X$. Assume that $\phi:X\rightarrow \mathbb{R}$ is Hölder continuous. R. Bowen shows that there is a unique equilibrium ...
4
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1answer
208 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 ...
1
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1answer
104 views

Continuity of Lyapunov spaces

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+\...
4
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2answers
343 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? ...
4
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3answers
360 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}^\...
6
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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 $T(I_i)=[0,1]$. ...
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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_{...
5
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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 ...
5
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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{...
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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=\...
7
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1answer
320 views

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 $G$ (...