## Search

## Analysis, Geometry, and Dynamical Systems

Inﬁnite alphabet ultragraph edge shift spaces: relations to $C^\ast$-algebras and chaos.

*Daniel Gonçalves*, Universidade Federal de Santa Catarina, Brasil.

## Abstract

We explain the notion of ultragraphs, which generalize directed graphs, and use this combinatorial object to deﬁne a notion of (one-sided) edge shift spaces (which, in the ﬁnite case, coincides with the edge shift space of a graph). We then go on to show that these shift spaces have some nice properties, as for example metrizability and basis of compact open sets. We examine shift morphisms between these shift spaces: we give an idea how to show that if two (possibly inﬁnite) ultragraphs have edge shifts that are conjugate, via a conjugacy that preserves length, then the associated ultragraph $C^\ast$-algebras are isomorphic. Finally we describe Li-Yorke chaoticity associated to these shifts and remark that the results obtained mimic the results for shifts of finite type over finite alphabets (what is not the case for infinite alphabet shift spaces with the product topology).

## Analysis, Geometry, and Dynamical Systems

Symbolic dynamics of piecewise contractions.

*Benito Frazão Pires*, Universidade de São Paulo.

## Abstract

A map $f:[0,1]\to [0,1]$ is a piecewise contraction if locally $f$ contracts distance, i.e., if there exist $0<\lambda<1$ and a partition of $[0,1]$ into intervals $I_1,I_2,\ldots,I_n$ such that $\left\vert f(x)-f(y)\right\vert \le\lambda \vert x-y\vert$ for all $x,y\in I_i$ $(1\le i\le n)$. Piecewise contractions describe the dynamics of many systems such as traffic control systems, queueing systems, outer billiards and Cherry flows. Here I am interested in the symbolic dynamics of such maps. More precisely, we say that an infinite word $i_0 i_1 i_2\ldots$ over the alphabet $\mathcal{A}=\{1,2,\ldots,n\}$ is the natural coding of $x\in [0,1]$ if $f^k(x)\in I_{i_k}$ for all $k\ge 0$. The aim of this talk is to provide a complete classification of the words that appear as natural codings of injective piecewise contractions.## Geometria em Lisboa

To be announced.

*Gonçalo Oliveira*, Universidade Federal Fluminense.

## Mini course on Hydrodynamic Limit of Particle Systems on Resistance Spaces

Mini-course on Hydrodynamic limit of particle systems on resistance spaces (I).

*Joe Chen*, Colgate University.

## Abstract

A main topic in probability theory is the study of scaling limits of random processes. One class of problems deals with scaling limits of single-particle Markov processes to a diffusion process. Another class of problems deals with scaling limits of many-particle Markov processes to a deterministic or stochastic differential equation. The former class has been studied on many state spaces, such as Euclidean spaces, manifolds, graphs, groups, etc. The latter class has been studied on Euclidean spaces, but not as much on non-Euclidean spaces.

The goal of my mini-course is to describe my recent progress on establishing scaling limits of many-particle systems on state spaces which are bounded in the resistance metric, a.k.a. *resistance spaces*. These include trees, fractals, and random graphs arising from critical percolation. As a concrete example, we can establish scaling limits of the weakly asymmetric exclusion process on the Sierpinski gasket interacting with 3 boundary reservoirs, which generalizes (in a nontrivial way) the analysis on the unit interval interacting with 2 boundary reservoirs. I will explain the key ideas behind these results, and discuss connections to the analysis of (S)PDEs, and issues of non-equilibrium statistical physics, on resistance spaces. From a technical point of view, I will address some novel functional inequalities for the exclusion process that relates to electrical resistance, and describe how they are used to effect *coarse-graining* in passing to the scaling limits.

## Mini course on Hydrodynamic Limit of Particle Systems on Resistance Spaces

Mini-course on Hydrodynamic limit of particle systems on resistance spaces (II).

*Joe Chen*, Colgate University.

## Abstract

A main topic in probability theory is the study of scaling limits of random processes. One class of problems deals with scaling limits of single-particle Markov processes to a diffusion process. Another class of problems deals with scaling limits of many-particle Markov processes to a deterministic or stochastic differential equation. The former class has been studied on many state spaces, such as Euclidean spaces, manifolds, graphs, groups, etc. The latter class has been studied on Euclidean spaces, but not as much on non-Euclidean spaces.

The goal of my mini-course is to describe my recent progress on establishing scaling limits of many-particle systems on state spaces which are bounded in the resistance metric, a.k.a. *resistance spaces*. These include trees, fractals, and random graphs arising from critical percolation. As a concrete example, we can establish scaling limits of the weakly asymmetric exclusion process on the Sierpinski gasket interacting with 3 boundary reservoirs, which generalizes (in a nontrivial way) the analysis on the unit interval interacting with 2 boundary reservoirs. I will explain the key ideas behind these results, and discuss connections to the analysis of (S)PDEs, and issues of non-equilibrium statistical physics, on resistance spaces. From a technical point of view, I will address some novel functional inequalities for the exclusion process that relates to electrical resistance, and describe how they are used to effect *coarse-graining* in passing to the scaling limits.

## Mini course on Hydrodynamic Limit of Particle Systems on Resistance Spaces

Mini-course on Hydrodynamic limit of particle systems on resistance spaces (III).

*Joe Chen*, Colgate University.

## Abstract

A main topic in probability theory is the study of scaling limits of random processes. One class of problems deals with scaling limits of single-particle Markov processes to a diffusion process. Another class of problems deals with scaling limits of many-particle Markov processes to a deterministic or stochastic differential equation. The former class has been studied on many state spaces, such as Euclidean spaces, manifolds, graphs, groups, etc. The latter class has been studied on Euclidean spaces, but not as much on non-Euclidean spaces.

The goal of my mini-course is to describe my recent progress on establishing scaling limits of many-particle systems on state spaces which are bounded in the resistance metric, a.k.a. *resistance spaces*. These include trees, fractals, and random graphs arising from critical percolation. As a concrete example, we can establish scaling limits of the weakly asymmetric exclusion process on the Sierpinski gasket interacting with 3 boundary reservoirs, which generalizes (in a nontrivial way) the analysis on the unit interval interacting with 2 boundary reservoirs. I will explain the key ideas behind these results, and discuss connections to the analysis of (S)PDEs, and issues of non-equilibrium statistical physics, on resistance spaces. From a technical point of view, I will address some novel functional inequalities for the exclusion process that relates to electrical resistance, and describe how they are used to effect *coarse-graining* in passing to the scaling limits.

## Mini course on Hydrodynamic Limit of Particle Systems on Resistance Spaces

Mini-course on Hydrodynamic limit of particle systems on resistance spaces (IV).

*Joe Chen*, Colgate University.

## Abstract

*resistance spaces*. These include trees, fractals, and random graphs arising from critical percolation. As a concrete example, we can establish scaling limits of the weakly asymmetric exclusion process on the Sierpinski gasket interacting with 3 boundary reservoirs, which generalizes (in a nontrivial way) the analysis on the unit interval interacting with 2 boundary reservoirs. I will explain the key ideas behind these results, and discuss connections to the analysis of (S)PDEs, and issues of non-equilibrium statistical physics, on resistance spaces. From a technical point of view, I will address some novel functional inequalities for the exclusion process that relates to electrical resistance, and describe how they are used to effect *coarse-graining* in passing to the scaling limits.

## Mini course on Hydrodynamic Limit of Particle Systems on Resistance Spaces

Hydrodynamic limit of particle systems on resistance spaces (V).

*Joe Chen*, Colgate University.

## Abstract

*resistance spaces*. These include trees, fractals, and random graphs arising from critical percolation. As a concrete example, we can establish scaling limits of the weakly asymmetric exclusion process on the Sierpinski gasket interacting with 3 boundary reservoirs, which generalizes (in a nontrivial way) the analysis on the unit interval interacting with 2 boundary reservoirs. I will explain the key ideas behind these results, and discuss connections to the analysis of (S)PDEs, and issues of non-equilibrium statistical physics, on resistance spaces. From a technical point of view, I will address some novel functional inequalities for the exclusion process that relates to electrical resistance, and describe how they are used to effect *coarse-graining* in passing to the scaling limits.

## Mini course on Hydrodynamic Limit of Particle Systems on Resistance Spaces

Hydrodynamic limit of particle systems on resistance spaces (VI).

*Joe Chen*, Colgate University.

## Abstract

*resistance spaces*. These include trees, fractals, and random graphs arising from critical percolation. As a concrete example, we can establish scaling limits of the weakly asymmetric exclusion process on the Sierpinski gasket interacting with 3 boundary reservoirs, which generalizes (in a nontrivial way) the analysis on the unit interval interacting with 2 boundary reservoirs. I will explain the key ideas behind these results, and discuss connections to the analysis of (S)PDEs, and issues of non-equilibrium statistical physics, on resistance spaces. From a technical point of view, I will address some novel functional inequalities for the exclusion process that relates to electrical resistance, and describe how they are used to effect *coarse-graining* in passing to the scaling limits.