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Thermodynamic formalism for intermittent maps with multiple neutral fixed points and phase transitions

This paper establishes the thermodynamic formalism for intermittent interval maps with multiple neutral fixed points and discontinuous potentials, proving the existence of a unique positive-entropy equilibrium state under hyperbolicity conditions while characterizing phase transitions where multiple ergodic states may coexist at critical parameters.

Original authors: Daniel Coronel, Francisco Monardes

Published 2026-08-14

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CC BY 4.0

Original authors: Daniel Coronel, Francisco Monardes

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine the universe as a giant, chaotic dance floor where particles, planets, and even ideas are constantly bumping into each other, spinning, and changing direction. In the world of physics and mathematics, scientists try to predict the "average" behavior of this chaos. They ask: If you watch this dance long enough, what does the crowd look like? Do they cluster in one corner, or do they spread out evenly? This field is called dynamical systems, and it's like trying to forecast the weather for a system that never stops moving.

To make sense of this chaos, mathematicians use a toolkit called thermodynamic formalism. Think of it as a way to assign a "score" or a "preference" to different ways the system can move. Just as a hot cup of coffee naturally cools down to reach a comfortable temperature, complex systems often settle into a specific, stable pattern called an equilibrium state. This is the "most likely" way the system behaves. Usually, if the system is very chaotic and spreads out quickly (like a gas in a room), there is only one perfect equilibrium state, and it's easy to predict. But what happens if the dance floor has sticky spots where dancers get stuck for a while before moving on? These "sticky spots" are called neutral fixed points, and they make the system behave strangely, sometimes creating multiple possible "most likely" patterns or causing the system to suddenly switch behaviors. This sudden switch is known as a phase transition, similar to water suddenly turning into ice.

In this paper, Daniel Coronel and Francisco Monardes tackle a tricky version of this dance floor: a map (a mathematical rule for moving points) that has multiple sticky spots (neutral fixed points) where things slow down, mixed with fast, chaotic areas. They want to know: When we change the "temperature" of the system (represented by a parameter called $\beta$ ), how does the equilibrium state change? Do we get one stable pattern, or do we get a messy mix of several?

The authors study a specific class of maps that stretch and fold an interval (like a piece of rubber) but have a few points where the stretching stops completely for a moment. They also look at "potentials," which are like rules that tell the system which paths are more desirable. Some of these rules can be a bit "jagged" or discontinuous, meaning the system might jump suddenly rather than sliding smoothly.

Here is what they found, broken down into the story of the dance floor:

The "Hyperbolic" Zone: One Perfect Dance
When the "temperature" is low enough (or the potential is strong enough in a specific way), the system behaves beautifully. The authors prove that there is exactly one equilibrium state. In this state, the system forgets its past very quickly (exponential decay of correlations), meaning if you nudge the dancers, they return to the average pattern almost immediately. It's a stable, predictable world where the math is clean and the system has a high "entropy" (a measure of how messy and diverse the movement is).

The Phase Transition: The Tipping Point
However, as they turn up the "temperature" (increasing the parameter $\beta$ ), the system hits a critical wall. The authors show that there is a specific critical value, let's call it $\beta^*$ , where things get weird.

Before the wall ( $\beta < \beta^*$ ): The system is still hyperbolic. There is one unique, chaotic, and positive-entropy dance.
At the wall ( $\beta = \beta^*$ ): The system hits a phase transition. Here, the rules change. The "pressure" (a mathematical score for the system's energy) stops curving and becomes a straight line. At this exact moment, the system might have multiple equilibrium states coexisting. One of these states is the usual chaotic dance with positive entropy, but the others are "stuck" on the sticky spots (the neutral fixed points).
After the wall ( $\beta > \beta^*$ ): The chaotic dance dies out. The system abandons the complex, spreading movement and settles entirely on the sticky spots. The only equilibrium states left are the ones where the dancers are just sitting on the neutral fixed points. The "pressure" function becomes a simple, straight line, indicating that the system has lost its complexity.

The "Sticky" vs. "Chaotic" Showdown
A key finding is that at the critical point ( $\beta^*$ ), you can have a mix, but it's limited. You might have one chaotic state with positive entropy, and a few other states where the system is just sitting on the sticky spots. But you can never have two different chaotic states with positive entropy at the same time. The paper proves that if a chaotic state exists, it is unique. If the system decides to sit on the sticky spots, it does so exclusively.

Why This Matters
The authors didn't just guess this; they built a rigorous mathematical proof using tools called conformal measures (which act like a special lens to see how the system stretches) and transfer operators (machines that calculate how the system evolves). They showed that even when the rules are a bit jagged or discontinuous, the system follows a strict logic: it either stays in a unique, chaotic, high-energy state, or it collapses into a few simple, low-energy states sitting on the sticky points.

They also clarified that this behavior is different from systems with only one sticky spot. With multiple sticky spots, the system has to choose which one to favor when it gets "cold" enough, leading to a richer set of possibilities for how the phase transition happens.

In short, the paper maps out the exact moment a complex, chaotic system loses its freedom and gets stuck, proving that while the transition can be messy, the rules governing the mess are surprisingly precise and predictable.

Technical Summary: Thermodynamic Formalism for Intermittent Maps with Multiple Neutral Fixed Points and Phase Transitions

Problem Statement
The paper addresses the thermodynamic formalism for a class of interval maps exhibiting intermittency, specifically those possessing finitely many neutral fixed points. While the thermodynamic formalism for uniformly hyperbolic systems is well-established—guaranteeing unique equilibrium states and real-analytic pressure functions for Hölder potentials—the situation for non-uniformly hyperbolic systems is significantly more complex. In systems with neutral fixed points (intermittency), such as the Manneville–Pomeau family, Hölder-continuous potentials may admit multiple equilibrium states, and the pressure function may fail to be real analytic, leading to phase transitions.

Existing literature has extensively studied maps with a single source of intermittency. However, maps with multiple neutral fixed points present additional challenges regarding the coexistence of uniformly expanding and intermittent branches, as well as the behavior of potentials near discontinuity points. This work extends the formalism to interval maps with finitely many branches and multiple neutral fixed points, considering a broad class of "branch Hölder-continuous" potentials that may be discontinuous at the partition boundaries.

Methodology
The authors employ a combination of geometric estimates, topological extensions, and functional analytic techniques:

Topological Extension: To handle the discontinuities inherent in the maps and potentials, the authors construct a continuous extension of the dynamical system. Following the procedure in [Kel98], they "double" the points in the preimage of the partition boundaries to create a compact metric space $X$ and a continuous map $\tilde{f}$ . This allows the application of standard results from the theory of continuous dynamical systems on compact spaces.
Conformal Measures: A central tool is the detailed analysis of conformal measures. The authors establish the existence of $\exp(P(\phi) - \phi)$ -conformal measures for the extended system and relate them back to the original system. They distinguish between atom-free conformal measures (associated with hyperbolic potentials) and atomic measures supported on neutral fixed points.
Transfer Operators and Banach Spaces: The study utilizes Keller spaces ( $H_{\gamma,1}(m)$ ), which are Banach spaces of functions with specific bounded variation and Hölder-like properties. The authors analyze the spectral properties of the transfer operator $\mathcal{L}_\phi$ acting on these spaces.
Hyperbolicity Condition: A key technical condition is the "hyperbolicity" of the potential, defined by the inequality $\sup_{x} \frac{1}{n} S_n \phi(x) < P(\phi)$ . The authors prove this is equivalent to the condition $P(\phi) > \max_{k \in N(f)} \phi(\xi_k)$ , where $\xi_k$ are the neutral fixed points.

Key Contributions and Results

Existence and Uniqueness under Hyperbolicity (Theorem B): For a subclass of branch Hölder-continuous potentials satisfying the hyperbolicity condition, the authors prove the existence of a unique equilibrium state with positive entropy. This state exhibits strong statistical properties, including exponential decay of correlations, and the pressure function is real analytic in a neighborhood of the potential. The transfer operator acting on the Keller space possesses a spectral gap.
Characterization of Conformal Measures (Theorem A): The paper provides a comprehensive description of conformal measures. It establishes that if a potential is hyperbolic, all conformal measures are atom-free. Conversely, if the hyperbolicity condition fails, atomic conformal measures may exist, specifically supported on maximizing neutral fixed points.
Phase Transitions (Theorem C): The main result characterizes the phase transitions for the one-parameter family $\beta \phi$ $β ϕ$ ( $\beta > 0$ $β > 0$ ). The authors prove a dichotomy:
1. The family remains hyperbolic for all $\beta > 0$ , retaining the properties of Theorem B.
2. There exists a critical parameter $\beta^*$ such that for $\beta \ge \beta^*$ , the pressure function becomes affine: $P(\beta \phi) = \beta \max_{k \in N(f)} \phi(\xi_k)$ .
  At the critical parameter $\beta^*$ , multiple ergodic equilibrium states may coexist. Specifically, there are at most $|N_{\max}(f)| + 1$ ergodic equilibrium states: $|N_{\max}(f)|$ are Dirac measures supported on maximizing neutral fixed points, and at most one has positive entropy and is absolutely continuous with respect to a fully supported conformal measure. For $\beta > \beta^*$ , the only equilibrium states are the Dirac measures on the maximizing neutral fixed points.
Geometric Potentials: The framework is designed to accommodate discontinuous potentials, allowing the results to apply to geometric potentials (e.g., $-\log |Df|$ ). The paper notes that a complete description of the phase transition for geometric potentials is provided in a companion paper [CM26].

Significance and Claims
The authors claim that their approach relies on a detailed analysis of conformal measures, which may be of independent interest. By allowing discontinuous potentials, the results apply to geometric potentials, a case of significant physical relevance.

The paper distinguishes itself from recent works on maps with multiple neutral fixed points (e.g., [CG26, CM25, CMT26, CL24, CLM23]) by requiring only $C^1$ regularity with Hölder-continuous derivatives (rather than $C^2$ ) and by allowing an arbitrary finite number of branches with potentially different degrees of "stickiness" at the neutral fixed points. However, the authors acknowledge a limitation: their approach requires the derivative to admit continuous extensions at discontinuity points and to be at least 1, a restriction not present in some recent works that allow derivatives to tend to zero or infinity.

The paper asserts that the results generalize known theorems for the Manneville–Pomeau family (single neutral fixed point) and Hölder-continuous potentials, while providing a new framework for the more complex setting of multiple neutral fixed points and discontinuous potentials. The proof of the "Key Lemma" (Lemma A.1), adapted from settings of rational and multimodal maps, is provided in the appendix as it does not directly apply to the current setting.

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