Questions tagged [thermodynamic-formalism]
Thermodynamic formalism is the study of equilibrium states, Gibbs measures and topological pressure for dynamical systems.
21
questions
2
votes
0
answers
318
views
A (possible) generic spectral property in one dimensional dynamics
Context and Definitions
Consider the interval $I=[0,1]$. We say that $T:I\to I$ satisfies the axiom A (I am following [1]) if:
$T$ has a finite number of hyperbolic periodic attractors; and
defining $...
- 233
2
votes
1
answer
270
views
Relationship between heat kernel and Maxwell-Boltzmann distribution
There appears to be a connection between the heat kernel and Maxwell-Boltzmann distribution, but I have not seen this in the literature before. I'd appreciate any kind comments or corrections/...
- 37
0
votes
1
answer
69
views
Seeking for references - Bowen Formula and a link between dimension theory and thermodynamic formalism
I'm needing references - preferably published papers and books - about this subject. I'm relatively new to the state of the art of fractal geometry and am way too inexperienced to seek for myself at ...
- 188
2
votes
0
answers
81
views
Biased ensemble in the unitary group
I am interested in studying the ensemble of unitary random matrices in $U(L)$ made as follows
$$
\mu(U)=\frac{1}{\mathcal{Z}[\omega]}\mu_{\rm Haar}(U) e^{-\sum_{k=1}^L \sum_{l=1}^N \omega_k |U_{kl}|^2}...
- 121
0
votes
0
answers
117
views
Generalized Ising Model
I am in very trouble with a particular expression. I leave the original pages in order to have everything available and what I am goin to leave are the first pages of nine chapter of Non Perturbative ...
1
vote
1
answer
107
views
Lyapunov spectrum($h_{\mathrm{top}}(K(\alpha))$) achieves a positive value somewhere
$\DeclareMathOperator{\top}{\mathrm{top}}$Let $(\Sigma, T)$ be a topologically mixing subshift of finite type and $f:\Sigma \to \mathbb{R}$ be a Hölder continuous map.
Let
$$K(\alpha)=\Big\{x\in \...
- 1,031
6
votes
2
answers
661
views
Explanation for why an ideal fluid doesn't have increasing entropy?
The equations of motion for a very simple ideal fluid (specifically a calorically perfect, monatomic, ideal gas) are \begin{align*}\dot{\rho}+\nabla \cdot (\rho u)=0 \;&\text{(mass conservation)} \...
- 51.6k
11
votes
0
answers
191
views
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: ...
- 60.4k
2
votes
0
answers
178
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 ...
8
votes
3
answers
1k
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 ...
3
votes
0
answers
92
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 ...
- 60.4k
4
votes
1
answer
311
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 ...
- 13.5k
1
vote
1
answer
122
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+\...
- 475
5
votes
3
answers
490
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}^\...
- 14.2k
4
votes
2
answers
372
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? ...
- 6,186
5
votes
1
answer
191
views
Multifractal Analysis and Dimension Spectrum of Unions
Consider the classical Multifractal Analysis, and the decomposition of the state space $X$ into level sets
$$X=\bigcup_{\alpha}\left\{x\mid d_\mu(x)=\alpha\right\}\cup\left\{x\mid d_\mu(x) \,\mathrm{...
CommunityBot
- 1
6
votes
1
answer
213
views
Estimates of Hausdorff dimension (and its derivatives)
For example, the cookie cutter maps, say $T:I_1 \cup I_2 \subset [0,1] \to [0,1] $ is a $C^2$ map such that $|T'|>1$ and provided $I_1$ and $I_2$ are disjoint closed intervals and $T(I_i)=[0,1]$. ...
- 1,898
2
votes
0
answers
126
views
Is $\text{Bow}(X,T)$ a Banach Space?
Let $X=\{0,1\}^{\mathbb{N}}$ be the sequence space and $T:X\to X$ the left shift mapping. Define the vector space $\text{Bow}(X,T)$ as
$$
\text{Bow}(X,T)=\{f\in C^{0}(X);~\sup_{n\in \mathbb{N}}\sup_{...
CommunityBot
- 1
5
votes
0
answers
223
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 ...
- 707
4
votes
2
answers
1k
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=\...
CommunityBot
- 1
7
votes
1
answer
355
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 $G$ (...
- 8,782